#  Inference on Extreme Quantiles of
Unobserved Individual Heterogeneity

Vladislav<EMAIL_ADDRESS>Universitat Pompeu Fabra and
Barcelona School of Economics†† Acknowledgments: I am indebted to Christian
Brownlees and Kirill Evdokimov for their support and guidance. I am also
grateful to Iván Fernández-Val, Adam Lee, Zoel Martin, Geert Mesters, Katerina
Petrova, Barbara Rossi, Alina Shirshikova, André Souza, Andrea Sy, Piotr
Zwiernik, and the participants at the 2021 European Winter Meeting of the
Econometric Society, the 2021 Spanish Economic Association Symposium, the 2022
Royal Economic Society Annual Conference, the 2022 BSE Summer Forum, the 27th
International Panel Data Conference, the 10th ICEEE, and the 3rd CESC for
comments and discussions.

( June 16, 2023 )

###### Abstract

We develop a methodology for conducting inference on extreme quantiles of
unobserved individual heterogeneity (heterogeneous coefficients, heterogeneous
treatment effects, etc.) in a panel data or meta-analysis setting. Inference
in such settings is challenging: only noisy estimates of unobserved
heterogeneity are available, and approximations based on the central limit
theorem work poorly for extreme quantiles. For this situation, under weak
assumptions we derive an extreme value theorem and intermediate order theorem
for noisy estimates and appropriate rate and moment conditions. Both theorems
are then used to construct confidence intervals for extremal quantiles. The
intervals are simple to construct and require no optimization. Inference based
on the intermediate order theorem involves a novel self-normalized
intermediate order theorem. In simulations, our extremal confidence intervals
have favorable coverage properties in the tail. Our methodology is illustrated
with an application to firm productivity in denser and less dense areas.

KEYWORDS: unobserved heterogeneity, panel data, heterogeneous coefficients,
estimation noise, extreme value theory, feasible inference

JEL CLASSIFICATION: C21, C23

###  1 Introduction

Extreme quantiles of unobserved individual heterogeneity (UIH) are often of
interest in analysis of economic panel data and in meta-analysis; UIH includes
heterogeneous coefficients, treatment effects, etc. For example, in the
setting of Combes et al. (2012) we might be interested in estimating the
lowest level of firm-specific productivity that is compatible with firm
survival in a given area. This corresponds to finding the zeroth quantile of
the distribution of productivity of surviving firms.222 Here and in the
empirical application, “productivity” is understood in the sense of firm-
specific total factor productivity in a Cobb-Douglas production function. This
view follows the literature on production function estimation (e.g. Ackerberg
et al. (2007)), rather than the literature on estimation of production
frontiers (e.g. Kumbhakar et al. (2020a, b)). In particular, each firm may be
viewed as always being on its firm-specific production frontier, and interest
centers on the heterogeneous parameters controlling the frontiers.

The key challenge to estimation and inference is that only noisy estimates of
UIH are available, and it is not a priori clear when such estimates yield
useful information about the quantiles of interest. These individual estimates
are constructed from individual time series, studies, or cluster-level data,
and thus are subject to estimation noise. This estimation noise and the true
UIH are often related in a complex and unobservable manner due to the
dependence between UIH and covariates used in estimation (e.g. Heckman (2001);
Browning and Carro (2007, 2010)). Importantly, estimation noise does not
“average out” when dealing with extreme quantiles, in contrast to estimation
of means.

In this paper we develop a methodology for conducting inference on extreme
quantiles of UIH on the basis of noisy estimates, and we establish sharp
conditions under which noisy estimates of UIH are informative about the
quantiles of interest. The noisy estimates can be used for estimation and
inference if the tails of their distribution converge to the tails of the
distribution of interest in a certain weak pointwise sense. We construct
confidence intervals for extreme quantiles of UIH using self-normalizing
ratios of extreme or intermediate order statistics of the estimates. We also
provide hypothesis tests about the support of UIH. To obtain the relevant
asymptotic distributions, we prove appropriate extreme and intermediate value
theorems (EVT and IVT) for noisy estimates.

The asymptotic distributions of the statistics we propose are based on extreme
and intermediate asymptotic approximations. Such approximations are suited to
inference on extreme quantiles, and complement the central approximations
constructed by Jochmans and Weidner (2022) for conducting inference on central
quantiles of UIH. The three asymptotic approximations differ in how the
quantile of interest is modeled as a function of the cross-sectional size.
Extreme order approximations model the quantile of interest as drifting to 0
or 1 at a rate proportional to the cross-sectional dimension. The resulting
confidence intervals and tests for extreme quantiles of UIH are based on a
ratio of extreme order statistics of the estimated UIH. The distribution of
such ratios depends on the extreme value (EV) index of the distribution of
latent true UIH, and thus is non-pivotal. The corresponding critical values
can be consistently estimated by subsampling. In intermediate order
approximations, the quantile of interest drifts to 0 or 1 at a rate slower
than the cross-sectional sample size. In this case we base confidence
intervals on a ratio statistic of intermediate order statistics. This ratio
exploits a deterministic difference between two perfectly dependent random
variables in order to normalize the statistic. This result may also be of
interest in the noiseless case. In contrast, central order approximations are
based on the central limit theorem for quantiles; the quantile of interest is
modeled as fixed.

We show that the noisy estimates of UIH can be used for estimation and valid
inference on the quantiles of interest under mild conditions. There are two
key restrictions. First, we impose weak assumptions on the marginal
distributions of UIH and of the estimation noise. We do not restrict the
dependence structure between the true UIH and the estimation noise. This
allows estimation noise and the true UIH to be related in a complex manner, as
they typically are outside of experimental settings (Heckman, 2001; Browning
and Carro, 2007, 2010). Second, the tail of the noisy distribution of
estimates has to converge to the tail of the latent distribution of interest
in a certain pointwise sense; a condition that we label pointwise asymptotic
tail equivalence.

When tail equivalence holds, noisy estimates inherit extreme value properties
of the UIH. This allows us to establish an EVT and supporting distributional
results for noisy estimates. These results in turn serve as a foundation for
inference.

For a broad class of distributions, we obtain sufficient conditions for the
tail equivalence conditions in terms of rate restrictions on the cross-
sectional dimensional and individual sample sizes. The sufficient conditions
only require typical moment or normality assumptions on the estimation noise
and a lower bound for the extreme value index. The rate restrictions are mild
if the distribution has an infinite right endpoint. For example, in this case
tail equivalence holds if $N/T^{4}\to 0$ if the estimation noise has more than
8 moments, where $N$ is the cross-section size and $T$ is the size of unit-
specific datasets. This restriction matches the condition for validity of
inference on central quantiles obtained by Jochmans and Weidner (2022).
Moreover, if the estimation noise is normal, $N$ may be almost exponentially
large relative to $T$. In contrast, for distributions with finite endpoints
the rate conditions may be more stringent. There are two main components which
make restrictions milder: how light the tails of the noise distribution are,
and how heavy the tails of the true distribution of the effects are.
Intuitively, lighter-tailed noise and heavier-tailed UIH both increase signal
strength.

In simulations we find that if rate conditions hold, extreme-order
approximations offer favorable coverage properties in the tails, including
when compared with other approximations. The rate conditions are important,
and their failure may lead to distorted inference.

Our methodology is illustrated with an application to firm productivity in
denser and less dense areas in the setting of Combes et al. (2012). We address
two key aspects of their analysis. First, we consider the issue of firm
selection. Firm selection is conjectured to left-truncate the productivity
distribution by creating a minimal productivity threshold required for firm
surival. We find no evidence of truncation in the productivity distributions
in both denser and less areas. This strengthens the previous finding of Combes
et al. (2012) that truncation must be the same in both kinds of areas, though
at an unknown level. Second, Combes et al. (2012) make the key assumption that
the productivity distributions in denser and less dense areas are the same up
to mean, variance, and truncation. We nonparametrically show that the tails of
the productivity distribution in denser and less dense areas are similar up to
mean and variance of the distribution, lending credence to that assumption.

This paper is related to several strands of literature. First, several recent
papers have proposed methods for studying distributional properties of UIH
(Evdokimov, 2010; Arellano and Bonhomme, 2012; Okui and Yanagi, 2019; Barras
et al., 2021; Jochmans and Weidner, 2022; Sasaki and Wang, 2022). There are
two key contributions related to our paper. Jochmans and Weidner (2022) study
the issue of estimating the distribution function and central quantiles of UIH
from noisy estimates. They develop rate conditions and asymptotic normality
results in a similar setting to ours. However, their methodology is based on a
central limit theorem for quantiles, which is not well-suited for inference on
extreme quantiles. Sasaki and Wang (2022) apply an extreme order approximation
they develop for noiseless data to extreme quantiles of coefficients in a
linear random coefficient model under a high level condition on the data. We
focus on the general problem of inference using noisy estimates that covers
linear, nonlinear, and nonparametric estimators and obtain sharp conditions
under which these estimates are informative. Second, extreme order
approximations are related to the fixed-$k$ tail inference methods of Müller
and Wang (2017) and Sasaki and Wang (2022), developed in a noiseless setting.
In contrast to Müller and Wang (2017), we propose optimization-free confidence
intervals that require no bounds on the EV index. Third, while our approach to
inference using intermediate order approximation is novel, methods using
intermediate order statistics are standard in the EV literature, and we refer
to ch. 4 of de Haan and Ferreira (2006) for textbook treatment. Last, our
setting and results should be contrasted with the literature on estimating
extreme conditional quantiles or treatment effects in cross-sectional settings
(Chernozhukov, 2005; Chernozhukov and Fernández-Val, 2011; Zhang, 2018). The
focus of those papers is on modeling extreme conditional quantiles of
observable quantities, whereas our focus lies squarely on unobserved
quantities and the issue of estimation noise.

The rest of the paper is organized as follows. Section 2 formalizes our setup
and assumptions. In section 3 we lay down the probabilistic foundations of our
inference theory by proving extreme and intermediate extreme value theorems
for noisy estimates. Building on these results, in section 4 we discuss three
approaches to inferences: extreme, intermediate, and central order asymptotic
approximations. Using those approximations we construct confidence intervals
and tests about the support of the distribution of unobserved UIH. In section
5 we explore performance of those CIs in a Monte Carlo setting. Section 6
contains the empirical application. All proofs are collected in the appendix.
We provide additional results in the online appendix, available from the
author’s website.

###  2 Setting and Assumptions

####  2.1 Problem statement

Suppose that unobserved individual heterogeneity $\theta_{i}$ is sampled in an
iid fashion from some distribution $F$. $\theta_{i}$ may be a treatment
effect, effect size, a coordinate of a vector of individual-specific
coefficients, value of a function at a point, etc. $i$ indexes cross-sectional
units or individual studies; in what follows we will refer to them as “units”.

Our goal is to estimate quantiles $F^{-1}(q)$ where $q$ is close to 0 or 1,
and to conduct inference on them. In particular, we are interested in
constructing confidence intervals and hypothesis tests for hypotheses like
$H_{0}:\theta_{i}\geq 0$, which is equivalent to $H_{0}:F^{-1}(0)\geq 0$.
Without loss of generality we focus on the right tail of $F$.

####  2.2 Data generating process

We do not observe $\theta_{i}$ directly; instead we only see noisy
observations $\vartheta_{i,T}$:

$\vartheta_{i,T}=\theta_{i}+\dfrac{1}{T_{i}^{p}}\varepsilon_{i,T_{i}},\quad
i=1,\dots,N,$ (1)

where $T_{i}$ is the sample size available for unit $i$,
$\varepsilon_{i,T_{i}}$ is the scaled estimation error,
$\varepsilon_{i,T}=O_{p}(1)$ for all $T$, and $p>0$ is the rate of convergence
of the estimator. For clarity of exposition, in what follows we assume that
for all units $T_{i}=T$ for some common $T$; in the online appendix we show
that our results naturally extend to the case where sample sizes are unequal
between units (or clusters). $p$ is determined by the estimation method used
in a given case. For estimators convergent at the parametric rate $T^{-1/2}$,
$p$ is equal to $1/2$, but we allow other rates.
$\operatorname{\mathbb{E}}[\varepsilon_{i,T}]$ may be nonzero and need not
converge to 0 as $T\to\infty$; however, this bias may not diverge as
$T\to\infty$.

Representation (1) is general. It is compatible with any estimation method for
which a suitable rate of convergence is known; $\varepsilon_{i,T}$ can always
be defined via the identity
$\varepsilon_{i,T}=T^{p}(\vartheta_{i,T}-\theta_{i})$. Intuitively,
$\vartheta_{i,T}$ are estimators of $\theta_{i}$ that have growing precision.
$\vartheta_{i,T}$ can potentially be biased. Our setting nests the setup
considered by Jochmans and Weidner (2022) and the typical setup of meta-
analysis (see e.g. Higgins et al. (2009)).

We provide several examples of how $\vartheta_{i,T}$ may be constructed.

######  Example 1 (Unit-wise OLS and IV).

Let $y_{it}=\theta_{i}x_{it}+u_{it}$. First suppose that
$\operatorname{\mathbb{E}}[u_{it}x_{it}]=0$. We estimate heterogeneous
coefficients $\theta_{i}$ by unit-wise OLS:
$\vartheta_{i,T}=(T^{-1}\sum_{t=1}^{T}x_{it}^{2})^{-1}T^{-1}\sum_{t=1}^{T}x_{it}y_{it}$.
Define
$\varepsilon_{i,T}=(T^{-1}\sum_{t=1}^{T}x_{it}^{2})^{-1}T^{-1/2}\sum_{t=1}^{T}x_{it}u_{it}$
to write $\vartheta_{i,T}$ as in eq. (1) for $p=1/2$. Now let $x_{it}$ be
endogenous in the sense that $\operatorname{\mathbb{E}}[u_{it}x_{it}]\neq 0$.
Suppose a valid instrument $z_{it}$ is available:
$\operatorname{\mathbb{E}}[z_{it}u_{it}]=0$,
$\operatorname{\mathbb{E}}[z_{it}x_{it}]\neq 0$. The unit-wise IV estimators
are given by
$\vartheta_{i,T}=(T^{-1}\sum_{t=1}^{T}z_{it}x_{it})^{-1}T^{-1}\sum_{t=1}^{T}z_{it}y_{it}$.
In this case
$\varepsilon_{i,T}=(T^{-1}\sum_{t=1}^{T}z_{it}x_{it})^{-1}T^{-1/2}\sum_{t=1}^{T}z_{it}u_{it}$.
Note that the distribution of $\varepsilon_{i,T}$ may be dissimilar to the
normal distribution even for large $T$ if $z_{it}$ is not a strong instrument.
Even if $u_{it}$ is normally distributed, the distribution of
$\varepsilon_{i,T}$ can be skewed, bimodal, and possess an infinite expected
value (Nelson and Startz, 1990).

######  Example 2 (Nonparametric regression).

Let $\theta_{i}\coloneqq\operatorname{\mathbb{E}}[y_{it}|x_{it}=x_{0}]$ for
some fixed value $x_{0}$. Let
$\vartheta_{i,T}=(\sum_{t=1}^{T}K\left({(x_{it}-x_{0})}/{h}\right))^{-1}{\sum_{t=1}^{T}y_{it}K({(x_{it}-x_{0})}/{h})}{}$
by the Nadaraya-Watson estimator of $\theta_{i}$ where $K$ is a kernel
function and $h$ is a bandwidth parameter. Let $u_{it}=y_{it}-\theta_{i}$, and
set
$\varepsilon_{i,T}=(\sum_{t=1}^{T}K({x_{it}-x_{0}}/{h}))^{-1}{\sqrt{Th}\sum_{t=1}^{T}u_{it}K({(x_{it}-x_{0})}/{h})}$.
Let $h=T^{-s},s\in(0,1)$. It holds that $\varepsilon_{i,T}=O_{p}(1)$ and
$\vartheta_{i,T}=\theta_{i}+T^{-(1-s)/2}\varepsilon_{i,T}$ under suitable
conditions on $h$. If $h$ is picked to optimize the convergence rate,
$\varepsilon_{i,T}$ has a non-zero mean even in the limit.

##### Notation

Let $\vartheta_{1,N,T}\leq\dots\leq\vartheta_{N,N,T}$ be the order statistics
of $\\{\vartheta_{1,T},\dots,\vartheta_{N,T}\\}$, and similarly let
$\theta_{1,N}\leq\dots\leq\theta_{N,N}$ be the order statistics of the latent
noiseless $\\{\theta_{1},\dots,\theta_{N}\\}$. “Noiseless maximum” always
stands for $\theta_{N,N}$; “noisy maximum” stands for $\vartheta_{N,N,T}$;
same applies to all other order statistics. $\Rightarrow$ denotes weak
convergence, both of random variables and functions.

####  2.3 Assumptions

######  Assumption 1.

For each $T$, $\\{(\theta_{i},\varepsilon_{i,T})\\}_{i=1,\dots,N}$ are
independent and identically distributed random vectors indexed by $i$.

Observations $\vartheta_{i,T}$ are sampled in an iid fashion. Note that
$\vartheta_{i,T}$ can be conditionally heteroskedastic:
$\operatorname{\mathrm{Var}}(\vartheta_{i,T}|\theta_{i})=\operatorname{\mathrm{Var}}(\varepsilon_{i,T}|\theta_{i})$
can depend on $\theta_{i}$, provided this variance exists.

######  Assumption 2.

The distribution $F$ of $\theta$ is in the weak domain of attraction of an
extreme value distribution with extreme value index
$\gamma\in\operatorname{\mathbb{R}}$.

Under assumption 2, the classical extreme value theorem of Gnedenko (1943)
applies to the latent noiseless distribution $F$. This will serve as the basis
for extending this extreme value convergence to the observed noisy data.
Without assumption 2, we would not be able to conduct inference using the
asymptotic behavior of the sample maximum even if we had access to the true
$\theta_{i}$. Assumption 2 is a mild assumption, satisfied by almost all
textbook continuous distributions and many discrete ones. Assumption 2 is
equivalent to certain regular variation conditions on $F$ (see theorem 1.2.1
in de Haan and Ferreira (2006) for precise conditions)

Two key features of our analysis are that we place no restrictions on the
dependence structure between $\theta_{i}$ and $\varepsilon_{i,T}$ and that we
impose no distributional assumptions on $\varepsilon_{i,T}$. This approach can
be motivated as follows. First, $\theta_{i}$ and $\varepsilon_{i,T}$ will
typically be related in a complex and unobservable manner outside of tightly
controlled experimental settings. Such dependence may arise if the agent
chooses some covariates with knowledge of $\theta_{i}$, and these covariates
are in turn used in estimation of $\theta_{i}$. Second, the distribution of
$\varepsilon_{i,T}$ might not be well-approximated by a normal distribution,
as in the IV case of example 1 (however, we show how to exploit normality of
$\varepsilon_{i,T}$ in establishing our rate conditions). In light of the
these challenges, we only impose a weak assumption on the marginal
distribution $G_{T}$ of $\varepsilon_{i,T}$.

######  Assumption 3.

$\\{G_{T}\\}$ form a tight family of distributions indexed by $T$.

Intuitively, assumption 3 requires that $\varepsilon_{i,T}$ be defined in such
a way that, as $T\to\infty$, the distributions $G_{T}$ of $\varepsilon_{i,T}$
do not escape to infinity. Assumption 3 holds automatically if
$T^{p}(\vartheta_{i,T}-\theta_{i})$ has a non-degenerate asymptotic
distribution. Together with definition (1), assumption 3 implies that each
$\vartheta_{i,T}$ is consistent for $\theta_{i}$, but we allow
$\vartheta_{i,T}$ to be biased in finite samples. In addition, assumption 3
allows the mean of $\varepsilon_{i,T}$ to be nonzero in the limit, as may
occur in nonparametric regression with MSE-minimizing choice of bandwidth
(example 2).

###  3 Extreme Value Theory For Noisy Estimates

####  3.1 Extreme Value Theorem For Noisy Estimates

The key first step towards conducting inference on extreme quantiles is to
establish distributional results for the sample maximum $\vartheta_{N,N,T}$.
The following theorem provides conditions under which $\vartheta_{N,N,T}$ and
the latent maximum $\theta_{N,N}=\max\\{\theta_{1},\dots,\theta_{N}\\}$ have
the same asymptotic distribution. It introduces the notion of tail equivalence
and serves as a stepping stone towards our inference results.

######  Theorem 3.1.

Let assumption 1 hold. Let constants $a_{N},b_{N}$ and a random variable $X$
be such that ${(\theta_{N,N}-b_{N})}/{a_{N}}\Rightarrow X$ as $N\to\infty$.
Let $F$ be the cdf of $\theta_{i}$ and $G_{T}$ be the cdf of
$\varepsilon_{i,T}$.

  1. (1)

_(Transferral of convergence)_ Let the following tail equivalence (TE)
conditions hold: for each $\tau\in(0,\infty)$ as $N,T\to\infty$ there exists
some $\epsilon\in(0,1)$

$\displaystyle\sup_{u\in\left[0,\epsilon\right]}\dfrac{1}{a_{N}}\left(F^{-1}\left(1-\dfrac{1}{N\tau}-u\right)-F^{-1}\left(1-\dfrac{1}{N\tau}\right)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(u\right)\right)\to
0,$ (TE-Sup)
$\displaystyle\inf_{u\in\left[0,\frac{1}{N\tau}\right]}\dfrac{1}{a_{N}}\left(F^{-1}\left(1-\dfrac{1}{N\tau}+u\right)-F^{-1}\left(1-\dfrac{1}{N\tau}\right)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(1-u\right)\right)\to
0.$ (TE-Inf)

Then $(\vartheta_{N,N,T}-b_{N})/{a_{N}}\Rightarrow X$ as $N,T\to\infty$.

  2. (2)

_(Sharpness)_ Consider the following TE condition: for each
$\tau\in(0,\infty)$ as $N,T\to\infty$

$\sup_{u\in\left[0,1-\frac{1}{N\tau}\right]}\dfrac{1}{a_{N}}\left(F^{-1}\left(1-\dfrac{1}{N\tau}-u\right)-F^{-1}\left(1-\dfrac{1}{N\tau}\right)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(u\right)\right)\to
0,$ (TE-Sup’)

Then (i) (TE-Sup) and (TE-Inf) together imply (TE-Sup’); (ii) (TE-Inf) and
(TE-Sup’) are sharp in the following sense: if at least one of the conditions
fails, there exists a sequence of joint distributions of
$(\theta_{i},\varepsilon_{i,T})$ with given marginal distributions $F,G_{T}$
such that $({\vartheta_{N,N,T}-b_{N}})/{a_{N}}$ weakly converges to a limit
different from $X$ or does not converge at all.

(TE-Sup) and (TE-Inf) are pointwise asymptotic tail equivalence conditions
between the distributions of $\vartheta_{i,T}$ and $\theta_{i}$. The
$F^{-1}+T^{-p}G^{-1}$ term approximately corresponds to the quantiles of the
noisy estimates, where the infimum and supremum adjust for the unknown joint
distribution of $\theta_{i}$ and $\varepsilon_{i,T}$. The $(-F^{-1})$ term
corresponds to the target noiseless quantiles. If (TE-Inf) and (TE-Sup) hold,
the distance between the tails of $F$ and the distribution of
$\vartheta_{i,T}$ is shrinking, pointwise in $\tau$. Observe that (TE-Sup) and
(TE-Inf) are trivially satisfied in a noiseless setting (that is, if
$G_{T}^{-1}=0$), as both the supremum and the infimum are attained at $u=0$
and are equal to zero.

Theorem 3.1 may be interpreted as a necessary and sufficient condition for
$\vartheta_{N,N,T}$ and $\theta_{N,N}$ to have the same asymptotic
distribution regardless of the dependence structure between $\theta_{i}$ and
$\varepsilon_{i,T}$. In particular, if conditions (TE-Sup’) and (TE-Inf) do
not hold, it is generally uncertain to what extent $\vartheta_{N,N,T}$
inherits the limit properties of $\theta_{N,N}$ (however, see remark 2 below
for a special case). For some joint distributions of
$(\theta_{i},\varepsilon_{i,T})$, $\vartheta_{N,N,T}$ might still have the
same asymptotic distribution as $\theta_{N,N}$. However, as the proof of
theorem 3.1 shows, there exists at least one possible asymptotic distribution
for $\vartheta_{N,N,T}$ that is different from the distribution of
$\theta_{N,N}$. Moreover, $\vartheta_{N,N,T}$ may only converge to it along a
subsequence and not overall.

We stress three important but subtle features of conditions (TE-Sup) and (TE-
Inf). First, the value of the infimum of (TE-Inf) is greater or equal to the
value of the supremum in (TE-Sup) (and (TE-Sup’)), as the proof shows. Second,
suppose that there is no estimation noise. Then the expression under the
supremum in (TE-Sup) is non-positive for all values of $u$ allowed and equal
to 0 at $u=0$. Similarly, the expression under the infimum in (TE-Inf) is non-
negative and also equal to zero at $u=0$. Similar logic persists in the
presence of estimation noise, though it affects the two conditions somewhat
asymmetrically. Suppose that $G_{T}(0)\in(0,1)$. Then the expression under the
infimum is eventually always non-negative due to the shrinking choice set for
$u$. (TE-Inf) requires the minimal value of that non-negative function to
converge to zero. Similarly, the expression under the supremum in (TE-Sup) is
negative for some interval of values of $u$, though it may change sign for
some larger values of $u$. Condition (TE-Sup) then requires than the supremum
of this function converges to 0, whether from above or below. We remark that
this is not a requirement of uniform convergence to 0 of the function under
the supremum. Last, to obtain sharpness it is necessary to consider (TE-Sup’),
which is weaker than (TE-Sup) in the presence of (TE-Inf), as we show in the
proof. However, for practical purposes (TE-Sup) is sufficient and easier to
interpret as controlling the contribution of the left tail of $G_{T}$.

######  Remark 1.

It is natural to use the same normalization constants $a_{N},b_{N}$ for both
$\vartheta_{N,N,T}$ and $\theta_{N,N}$. The distribution of $\vartheta_{i,T}$
is composed of two components: the $G_{T}$ component that decays as
$T\to\infty$, and the $F$ component. As the $F$ component does not decay, it
is appropriate to use the same $a_{N}$. For inference, the centering constants
$b_{N}$ will be quantiles of $F$ (see lemma 4.1 below). Since they are the
primary object of interest, it is natural to center at them.

######  Remark 2 (Knife-edge failure of (TE-Inf) and (TE-Sup)).

A special knife-edge case arises if for each $\tau\in(0,\infty)$ the
expressions in (TE-Inf) and (TE-Sup) (or (TE-Sup’)) converge to the same
$\delta_{\tau}$. In this case $a_{N}^{-1}\left(\vartheta_{N,N,T}-b_{N}\right)$
does converge. However, if $\delta_{\tau}$ is not a constant function of
$\tau\in(0,\infty)$, this limit distribution has a shape different from the
limit of $a_{N}^{-1}(\theta_{N,N}-b_{N})$. Correspondingly, it is not
necessarily a generalized extreme value distribution. We conjecture that
inference is still possible in this case using a suitable modification of
theorems 4.3-4.4 below.

When do conditions (TE-Inf) and (TE-Sup) hold? Sufficient conditions for (TE-
Inf) and (TE-Sup) take the form of restrictions on the magnitude of $N$
relative to $T$, given assumptions on the tails of $G_{T}$. Such conditions
depend on $\gamma$, the extreme value index of $F$, which is unknown. However,
a sufficient condition is possible if a lower bound $\gamma^{\prime}$ for
$\gamma$ is available. This is intuitive. Distributions with larger values of
$\gamma$ have heavier and longer tails. This heavier tail of $F$ will be more
pronounced in the data relative to the contribution of the noise, and so a
larger value of $N$ will be allowable. We state these conditions formally:

######  Proposition 3.2.

Let assumptions 1 and 3 hold. Let one of the following conditions hold:

  1. (1)

Let
$\sup_{T}\operatorname{\mathbb{E}}\lvert\varepsilon_{i,T}\rvert^{\beta}<\infty$
for some $\beta>0$, and let
${N^{1/\beta-\gamma^{\prime}}(\log(T))^{1/\beta}}/{T^{p}}\to 0$ for some
$\gamma^{\prime}$.

  2. (2)

For all $T$, let $\varepsilon_{i,T}\sim N(\mu_{T},\sigma_{T}^{2})$, and let
${N^{-\gamma^{\prime}}\sqrt{\log(N)}}/{T^{p}}\to 0$ for some
$\gamma^{\prime}$.

In addition, let $F$ satisfy assumption 2 with EV index
$\gamma>\gamma^{\prime}$. Then (TE-Inf) and (TE-Sup) hold for $F$ and $G_{T}$
for any sequence $\\{(a_{N},b_{N})\\}_{N=1}^{\infty}$ such that
$(\theta_{N,N}-b_{N})/a_{N}$ converges to a non-degenerate random variable.

As proposition 3.2 shows, two key factors determine how stringent the rate
restrictions are. First, as noted above, a heavier right tail of $F$ (larger
$\gamma^{\prime}$) allows a larger value of $N$ relative to $T$. Second,
lighter tails of $G_{T}$ lead to milder restrictions on $N$. The conditions on
tail of $G_{T}$ are captured by (1) and (2). Condition (1) only requires
$G_{T}$ to have uniformly bounded $\beta$th moments; (2) is a stronger
assumption of exact normality, it may be seen as a limiting version of (1) as
$\beta\to\infty$. In practice, we expect $G_{T}$ to lie between the two
extremes.

It is useful to compare the rate conditions of proposition 3.2 to the
conditions obtained by Jochmans and Weidner (2022) for inference on central
quantiles. Jochmans and Weidner (2022) show that $N/T^{4}\to 0$ is sufficient
for valid inference for a broad class of distributions $F$ if $p=1/2$ (see
theorem 4.6 below). The same rate condition arises in our extreme setting if
$F$ has an infinite tail $(\gamma\geq 0)$ and $G_{T}$ has more than 8 moments
(see remark 3 below for importance of $\gamma\geq 0$).333It is fairly
straightforward to impose conditions under which $G_{T}$ has a given number of
moments for linear estimators, such as those of examples 1 and 2. Further,
Brownlees and Morozov (2022) provide some conditions on existence of moments
of estimation error for nonlinear estimators. Further, proposition 3.2 allows
$N$ to be almost exponentially large compared to $T$ if $\varepsilon_{i,T}$ is
normally distributed. In such settings tail equivalence may hold even if the
conditions for validity of inference on central quantiles do not. If
$\gamma<0$, the opposite may happen, and the conditions of proposition 3.2 may
be more restrictive. We further highlight that our tail equivalence conditions
and the conditions of Jochmans and Weidner (2022) reflect two fundamentally
different aspects of the estimation problem. The conditions of Jochmans and
Weidner (2022) are driven by the order of the bias incurred by estimation. In
contrast, the tail equivalence conditions control the balance between the
right tail of the distribution of interest $F$ and (both) tails of the
distribution $G_{T}$ of estimation noise. Correspondingly, these conditions
benefit if the tails of $G_{T}$ are lighter; in particular, the conditions
hold trivially in a noiseless setting, as we note after theorem 3.1.

######  Remark 3.

An appropriate value of $\gamma^{\prime}$ might often be apparent in a given
application. For example, Gabaix (2009, 2016) documents that many economic
relations follow a power law and outlines some general theoretical mechanisms
under which a power law arises. If such a mechanism is likely to hold in a
given situation, it is reasonable to assume that the distribution of the data
is well approximated by a power law. Since a power law distribution has to
have $\gamma>0$, it is sufficient to check the hypothesis of proposition 3.2
with $\gamma^{\prime}=0$. Further, to allow the distribution of $\theta$ to
potentially have a light infinite tail, it is sufficient to check conditions
for any $\gamma^{\prime}<0$, potentially arbitrarily close to 0.

####  3.2 Intermediate Order Statistics

To conduct inference on extreme quantiles, we also need to develop an
asymptotic theory for intermediate order statistics. Formally,
$\theta_{N-k(N),N}$ is called an intermediate order statistic if
$k(N)\to\infty$ as $N\to\infty$ and $k(N)=o(N)$. Intuitively, such statistics
asymptotically stay in the tail, but are not the top statistics. We generally
suppress dependence of $k$ on $N$.

To derive asymptotic properties of intermediate order statistics, we impose an
additional assumption on $F$ that refines assumption 2.

######  Assumption 4 ($F$ satisfies a first order von Mises condition).

_$F$ is twice differentiable with density $f$, $f$ positive in some left
neighborhood of $F^{-1}(1)$ ($F^{-1}(1)$ might be finite or infinite), and for
some $\gamma\in\operatorname{\mathbb{R}}$ it holds that $\lim_{t\uparrow
F^{-1}(1)}\left([1-F]/{f}\right)^{\prime}(t)=\gamma.$ _

Assumption 4 is a slight strengthening of assumption 2. See Dekkers and de
Haan (1989) and de Haan and Ferreira (2006) for a discussion of the condition
and its plausibility.

We now state a theorem describing the asymptotic behavior of intermediate
order statistics:

######  Theorem 3.3 (Intermediate value theorem (IVT)).

Let assumptions 1 and 4 hold and let $k=k(N)\to\infty,k=o(N)$ as $N\to\infty$.
Define $c_{N}$ as the derivative of the inverse of $1/(1-F)$ evaluated at
$N/k$ and multiplied by $N/k$, that is,
$c_{N}=(N/k)\times\left[\left(\left({1}/({1-F})\right)^{-1}\right)^{\prime}\left(N/k\right)\right]$.
Let $U_{1},\dots,U_{N}$ be iid Uniform[0, 1] random variables.

  1. (1)

_(Transferral of convergence)_ Let the following tail equivalence conditions
hold as $N,T\to\infty$ for some $\epsilon>0$:

$\displaystyle\sup_{u\in\left[0,\epsilon\right]}\dfrac{\sqrt{k}}{c_{N}}\left(F^{-1}(1-U_{k,N}-u)-F^{-1}\left(1-U_{k,N}\right)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(u\right)\right)\xrightarrow{p}0,$
(2)
$\displaystyle\inf_{u\in\left[0,U_{k,N}\right]}\dfrac{\sqrt{k}}{c_{N}}\left(F^{-1}\left(1-U_{k,N}+u\right)-F^{-1}\left(1-U_{k,N}\right)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(1-u\right)\right)\xrightarrow{p}0.$
(3)

Then as $N,T\to\infty$

$\sqrt{k}\dfrac{\vartheta_{N-k,N,T}-F^{-1}\left(1-k/N\right)}{c_{N}}\Rightarrow
N(0,1).$ (4)

  2. (2)

_(Sharpness)_ Consider the following condition:

$\sup_{u\in\left[0,1-U_{k,N}\right]}\dfrac{\sqrt{k}}{c_{N}}\left(F^{-1}(1-U_{k,N}-u)-F^{-1}\left(1-U_{k,N}\right)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(u\right)\right)\xrightarrow{p}0,$
(5)

Conditions (3) and (5) are sharp in the sense of theorem 3.1.

Theorem 3.3 is the intermediate order counterpart of theorem 3.1; conditions
(2), (3), and (5) are analogs of conditions (TE-Sup), (TE-Inf), and (TE-Sup’),
respectively. The key difference between the two sets of conditions is the
region where tail equivalence is imposed. Theorem 3.1 concerns quantiles of
the form $1-1/(N\tau)$, $\tau>0$ fixed, whereas theorem 3.3 looks at quantiles
of the order $1-k/N$ (as we show in the proofs, $U_{k,N}=O_{p}(k/N)$). Since
$k\to\infty$, the two regions are asymptotically distinct. A second (minor)
point of difference between the two pairs of conditions is that conditions of
theorem 3.3 take a randomized form. As we show in proposition 3.4 below, this
randomness is easy to control, giving deterministic sufficient rate
conditions.

As for (TE-Inf) and (TE-Sup), sufficient conditions for (2) and (3) take form
of rate restrictions on $N$ and $T$ that depend on the EV index $\gamma$. A
sufficient condition is possible if a lower bound for $\gamma$ is available.
The following proposition is an analog of proposition 3.2.

######  Proposition 3.4.

Let assumptions 1 and 3 hold. Let $\delta\in(0,1)$. Let one of the following
conditions hold

  1. (1)

$\sup_{T}\operatorname{\mathbb{E}}\lvert\varepsilon_{i,T}\rvert^{\beta}<\infty$
and for some $\nu>0,\gamma^{\prime}$ it holds that
${N^{\delta/2(1+1/\beta)+(1-\delta)(-\gamma^{\prime}+1/\beta)+\nu/\beta}}/{T^{p}}\to
0$

  2. (2)

For all $T$ let $\varepsilon_{i,T}\sim N(\mu_{T},\sigma_{T}^{2})$, and for
some $\gamma^{\prime}$ it holds that
${N^{\delta/2+(1-\delta)(-\gamma^{\prime})}\sqrt{\log(N)}}/{T^{p}}\to 0$.

In addition, let $F$ satisfy assumption 4 with EV index
$\gamma>\gamma^{\prime}$. Then conditions (2) and (3) hold for $F$ and $G_{T}$
for $k=N^{\delta}$.

######  Remark 4 (Comparison of sufficient conditions for the EVT and the
IVT).

Depending on $\gamma$, the rate conditions for the EVT may be more or less
restrictive than the conditions for the IVT. For example, suppose that
$\varepsilon_{i\,T}$ is normally distributed. In this case for the EVT to hold
it is sufficient that ${N^{-\gamma}\sqrt{\log(N)}}/{T^{p}}\to 0$; for the IVT
to hold it is sufficient that
${N^{\delta/2+(1-\delta)(-\gamma)}\sqrt{\log(N)}}/{T^{p}}\to 0$. If
$\gamma\leq-1/2$, then the EVT condition implies the IVT condition; the
opposite holds if $\gamma>-1/2$, regardless of the value of $\delta$. In
particular, if $\gamma>0$, for the EVT there are no restrictions on relative
sizes of $N$ and $T$, but there are restrictions for the IVT. 444 A somewhat
related condition is obtained by Girard et al. (2021) in the context of
estimating expectiles in a regression model with heavy-tailed noise in a
cross-sectional setting. The key condition of their paper is that the
estimation noise is uniformly small (see their condition (2)). This condition
is naturally sufficient for theorem 3.3 (modulo the different normalizing
factors involved). In contrast, the conditions of theorem 3.3 only target the
differences between intermediate order statistics, and do not restrict the
difference between central or top order statistics.

######  Remark 5 (Dependence on $\delta$).

Conditions of proposition 3.4 depend on $\delta$, the parameter that
determines the magnitude of $k=N^{\delta}$. If $\delta$ is close to zero,
conditions for the IVT are close to those for the EVT. Intuitively, in this
case conditions (2) and (3) require asymptotic tail equivalence in
approximately the same section of the tail as conditions (TE-Sup) and (TE-
Inf), and so the resulting sufficient conditions are similar. As $\delta$
grows, the region controlled by conditions (2) and (3) becomes distinct from
the right endpoint of the distribution (while still staying in the tails by
requirement that $k=o(N)$).

###  4 Inference Using Noisy Estimates

We now turn to inference. In this section, we introduce confidence intervals,
estimators, and tests based on extreme and intermediate order asymptotic
approximations. We also briefly discuss the central order approximations of
Jochmans and Weidner (2022).

The extreme, intermediate, and central order approximations differ in their
appropriate use case. First, extreme and intermediate order approximations
should be employed when there are relatively few data points to the right of
the corresponding sample quantile. In this case the central limit theorem for
quantiles is unlikely to provide a good approximation, and central order
approximations should be not be used. Second, the choice between extreme and
intermediate order approximations is further determined by the magnitude of
$N$. Extreme order approximations are applicable even for $N$ fairly small, in
addition to having certain desirable robustness properties, as shown by Müller
and Wang (2017). In contrast, intermediate order approximations require a
fairly large value of $N$. Such approximations are controlled by the parameter
$k$ which is required to satisfy two conditions: $k$ must be large, while
$k/N$ must be small. These conditions may be incompatible in smaller datasets.

In addition, the three approximations differ in how the quantile of interest
is modeled. This choice determines the resulting limit distributions, whose
critical values are used to form the confidence intervals of interest or to
provide corrected estimators for quantiles.

####  4.1 Inference Using Extreme Order Approximations

Extreme order approximations use theorem 3.1 as the basis for inference. The
quantile of interest is modeled as drifting to 1 at a rate proportional to
$N^{-1}$. Formally, we select $b_{N}=F^{-1}(1-l/N)$ in theorem 3.1 for some
fixed $l>0$. Further, we make an explicit choice of scaling constants $a_{N}$.
Then the following version of theorem 3.1 holds:

######  Lemma 4.1.

Let assumptions of theorem 3.1 hold. Let $l>0$ be fixed. Let $E_{1}^{*}$ be a
standard exponential random variable.

  1. (1)

If $F$ satisfies assumption 2 with EV index $\gamma>0$, then

$\dfrac{1}{F^{-1}\left(1-\frac{1}{N}\right)}\left[\vartheta_{N,N,T}-F^{-1}\left(1-\dfrac{l}{N}\right)\right]\Rightarrow(E_{1}^{*})^{-\gamma}-l^{-\gamma}\text{
as }N,T\to\infty.$ (6)

  2. (2)

If $F$ satisfies assumption 2 with EV index $\gamma=0$, there for some
positive function $\hat{f}$

$\dfrac{1}{\hat{f}\left(F^{-1}\left(1-\frac{1}{N}\right)\right)}\left(\vartheta_{N,N,T}-F^{-1}\left(1-\frac{l}{N}\right)\right)\Rightarrow-\log(E_{1}^{*})-\log(l)\text{
as }N,T\to\infty.$ (7)

  3. (3)

If $F$ satisfies assumption 2 with EV index $\gamma<0$, then
$F^{-1}(1)<\infty$ and

$\dfrac{1}{F^{-1}(1)-F^{-1}\left(1-\frac{1}{N}\right)}\left(\vartheta_{N,N,T}-F^{-1}\left(1-\frac{l}{N}\right)\right)\Rightarrow-(E_{1}^{*})^{-\gamma}+l^{-\gamma}\text{
as }N,T\to\infty.$ (8)

Lemma 4.1 cannot be used for inference directly, as the scaling constants
involved are unknown. These constants involve the $(1-1/N)$th quantile of $F$,
and so are not covered by sample information.

To address this challenge, we first establish an intermediate result. The
following lemma extends theorem 3.1 to cover a vector of $q$ top order
statistics for $q$ fixed.

######  Lemma 4.2 (Joint EVT).

Let assumptions of theorem 3.1 hold. Let $q$ be a fixed natural number and
$E_{1}^{*},\dots,E_{q+1}^{*}$ be iid standard exponential random variables.

  1. (1)

If $F$ satisfies assumption 2 with $\gamma>0$, then as $N,T\to\infty$

$\begin{pmatrix}\dfrac{\vartheta_{N,N,T}}{F^{-1}(1-1/N)},\dfrac{\vartheta_{N-1,N,T}}{F^{-1}(1-1/N)},\dots,\dfrac{\vartheta_{N-q,N,T}}{F^{-1}(1-1/N)}\end{pmatrix}\\\
\Rightarrow\begin{pmatrix}{(E_{1}^{*})^{-\gamma}},{(E_{1}^{*}+E_{2}^{*})^{-\gamma}},\dots,{(E_{1}^{*}+E_{2}^{*}+\dots+E^{*}_{q+1})^{-\gamma}}\end{pmatrix}.$
(9)

  2. (2)

If $F$ satisfies assumption 2 with $\gamma=0$, then as $N,T\to\infty$ for
$\hat{f}$ as in lemma 4.1.

$\begin{pmatrix}\dfrac{\vartheta_{N,N,T}-F^{-1}\left(1-\frac{1}{N}\right)}{\hat{f}\left(F^{-1}\left(1-\frac{1}{N}\right)\right)},\dfrac{\vartheta_{N-1,N,T}-F^{-1}\left(1-\frac{1}{N}\right)}{\hat{f}\left(F^{-1}\left(1-\frac{1}{N}\right)\right)},\dots,\dfrac{\vartheta_{N-q,N,T}-F^{-1}\left(1-\frac{1}{N}\right)}{\hat{f}\left(F^{-1}\left(1-\frac{1}{N}\right)\right)}\end{pmatrix}\\\
\Rightarrow\begin{pmatrix}{-\log(E_{1}^{*})},{-\log(E_{1}^{*}+E_{2}^{*})},\dots,{-\log(E_{1}^{*}+E_{2}^{*}+\dots+E^{*}_{q+1})}\end{pmatrix}.$
(10)

  3. (3)

If $F$ satisfies assumption 2 with EV index $\gamma<0$, then as $N,T\to\infty$

$\begin{pmatrix}\dfrac{\vartheta_{N,N,T}-F^{-1}(1)}{F^{-1}(1)-F^{-1}(1-1/N)},\dfrac{\vartheta_{N-1,N,T}-F^{-1}(1)}{F^{-1}(1)-F^{-1}(1-1/N)},\dots,\dfrac{\vartheta_{N-q,N,T}-F^{-1}(1)}{F^{-1}(1)-F^{-1}(1-1/N)}\end{pmatrix}\\\
\Rightarrow\begin{pmatrix}{-(E_{1}^{*})^{-\gamma}},-{(E_{1}^{*}+E_{2}^{*})^{-\gamma}},\dots,-{(E_{1}^{*}+E_{2}^{*}+\dots+E^{*}_{q+1})^{-\gamma}}\end{pmatrix}.$
(11)

Lemma 4.2 allows us to solve the issue of unknown scaling rates by using an
observation due to Chernozhukov and Fernández-Val (2011). By taking the ratio
of two elements in the joint EVT 4.2, we eliminate scaling factors completely,
while the form of the limit is explicitly known up to the EV index $\gamma$.

Combining lemmas 4.1 and 4.2, we obtain the following version of the EVT that
can be used to conduct inference on extreme quantiles under tail equivalence
conditions:

######  Theorem 4.3 (Feasible EVT).

Let assumptions of theorem 3.1 hold, in particular, let $F$ have EV index
$\gamma\in\operatorname{\mathbb{R}}$. Let $q\geq 1,r\geq 0$ be fixed integers
and $l>0$ be a fixed real number; let $E_{1}^{*},E_{2}^{*},\dots$ be iid
standard exponential RVs. Then as $N,T\to\infty$

$\displaystyle\dfrac{\vartheta_{N-r,N,T}-F^{-1}\left(1-\frac{l}{N}\right)}{\vartheta_{N-q,N,T}-\vartheta_{N,N,T}}$
$\displaystyle\Rightarrow\dfrac{(E_{1}^{*}+\dots+E_{r+1}^{*})^{-\gamma}-l^{-\gamma}}{(E_{1}^{*}+\dots+E_{q+1}^{*})^{-\gamma}-(E_{1}^{*})^{-\gamma}},$
(12)

where for $\gamma=0$ the right hand side means
$[\log(E_{1}^{*})-\log(l)]/[\log(E_{1}^{*}+\dots+E_{q+1}^{*})-\log(E_{1}^{*})]$.
In addition, if $F$ satisfies assumption 2 with $\gamma<0$, then
$F^{-1}(1)<\infty$ and we may set $l=0$ in eq. (12).

Theorem 4.3 allows us to conduct inference on extreme quantiles with no
knowledge of the value of $\gamma$. The left hand side of eq. (12) does not
depend on $\gamma$; this statistic is the basis of our inference procedures.
While the right hand side limit distribution of eq. (12) is non-pivotal and
does depend on $\gamma$, we show below how the critical values can be
consistently estimated by subsampling without estimating $\gamma$ (while
$\gamma$ may be estimated consistently in our setting, we recommend against it
unless $N$ is fairly large, see remark 9 below).

We show how to construct confidence intervals, estimators and hypothesis tests
for extreme quantiles based on theorem 4.3 with a series of examples.

######  Example 3 (Location-scale equivariant confidence interval for 95th
percentile).

Let $l=10$ and $N=200$, in which case $F^{-1}(1-l/N)=F^{-1}(0.95)$. There are
only 10 observations to the right of the sample quantile, and it is
appropriate to use extreme order approximation described by theorem 4.3. Let
$q\geq 1$, $r\geq 0$ be fixed integers, see remark 6 below on choice of $r$
and $q$. Let $\hat{c}_{\alpha}$ be a consistent estimator of the $\alpha$th
quantile of
$\left[(E_{1}^{*}+\dots+E_{r+1}^{*})^{-\gamma}-10^{-\gamma}\right]/\left[(E_{1}^{*}+\dots+E_{q+1}^{*})^{-\gamma}-(E_{1}^{*})^{-\gamma}\right]$.
Then let the confidence interval for $F^{-1}(0.95)$ based on sample size
$N=200$ be given by:

$CI_{\alpha}=\left[\vartheta_{N-r,N,T}-\hat{c}_{\alpha/2}\left(\vartheta_{N-q,N,T}-\vartheta_{N,N,T}\right),\vartheta_{N-r,N,T}-\hat{c}_{1-\alpha/2}\left(\vartheta_{N-q,N,T}-\vartheta_{N,N,T}\right)\right].$
(13)

$CI_{\alpha}$ is location-scale equivariant, as the statistic of eq. (12) is
location-scale invariant.

Care must be exercised in interpreting the asymptotic properties of
$CI_{\alpha}$: it is a $(1-\alpha)\times 100\%$ asymptotic confidence
intervals for $F^{-1}\left(1-l/N\right)$. The target quantity shifts with $N$,
and the value of $l$ determines which quantile is targeted for a given sample
size.

######  Example 4 (Median-unbiased estimator for 95th percentile).

Let $q,r$, and $\hat{c}_{\alpha}$ be as in example (3). By theorem 4.3
$P\left(({\vartheta_{N-r,N,T}-F^{-1}(1-l/N)})/({\vartheta_{N-q,N,T}-\vartheta_{N,N,T}})\leq\hat{c}_{1/2}\right)\to
1/2$. Rearranging, we obtain that the estimator

$\mathcal{M}_{N,T}=\vartheta_{N-r,N,T}-\hat{c}_{1/2}(\vartheta_{N-q,N,T}-\vartheta_{N,N,T})$
(14)

is asymptotically median-unbiased for $F^{-1}(1-l/N)$ (see Chernozhukov and
Fernández-Val (2011) for a similar construction in a quantile regression
setting). Note that $\mathcal{M}_{N,T}$ is always contained in $CI_{\alpha}$
(unlike $\vartheta_{N-r,N,T}$ which lies in $CI_{\alpha}$ if
$\hat{c}_{\alpha/2}$ and $\hat{c}_{1-\alpha/2}$ have different signs).

######  Example 5 (Hypothesis tests about support).

We can also use theorem 4.3 to test hypotheses about the support of $F$. Let
$\gamma<0$ and suppose we wish to test $H_{0}:F^{-1}(1)\leq C$ vs.
$H_{1}:F^{-1}(1)>C$. Define the test statistic
$W_{C}={(\vartheta_{N,N,T}-C)}/{(\vartheta_{N-q,N,T}-\vartheta_{N,N,T})}.$ The
test rejects $H_{0}$ if $W_{C}<\hat{c}_{\alpha}$ where $\hat{c}_{\alpha}$ is a
consistent estimator of the $\alpha$th quantile of
${(E_{1}^{*})^{-\gamma}}/[{(E_{1}^{*}+\dots+E_{q+1}^{*})^{-\gamma}-(E_{1}^{*})^{-\gamma}}]$.
The test is asymptotically size $\alpha$ and consistent against point
alternatives, since $P(W_{C}<\hat{c}_{\alpha}|F^{-1}(1)=C)\to\alpha$, and for
any $\delta>0$, $P(W_{C}<\hat{c}_{\alpha}|F^{-1}(1)=C-\delta)\to 0$,
$P(W_{C}<\hat{c}_{\alpha}|F^{-1}(1)=C+\delta)\to 1$.

######  Remark 6 (Choice of $r$ and $q$).

Constructing CIs, median-unbiased estimators, and tests based on eq. (12)
requires choosing the tuning parameters $r$ and $q$. There are two leading
choices for $r$: $r=\lfloor l\rfloor$ and $r=0$, all other values are
dominated by these two in simulations. Confidence intervals based on
$r=\lfloor l\rfloor$ are centered at the sample $(1-l/N)$th quantile; those
based on $r=0$ are centered at the sample maximum. Similarly, the median-
unbiased estimator $\mathcal{M}_{N,T}$ of eq. (14) is based on adjusting
either the corresponding sample quantile or the sample maximum depending on
whether $r=\lfloor l\rfloor$ or $r=0$. In simulations of section 5 and the
online appendix we show that both options perform favorably when constructing
CIs. The intervals with $r=\lfloor l\rfloor$ display better length properties,
while the intervals with $r=0$ have slightly better coverage properties for
less extreme quantiles. In contrast, only $\mathcal{M}_{N,T}$ with $r=\lfloor
l\rfloor$ offers an improvement over the raw sample quantile in terms of mean
absolute error. The parameter $q$ can be any positive integer. We recommend
picking $q$ in the range 2-10 based on unreported simulation evidence. Picking
$q$ in this range appears to not affect performance of the CIs and the
estimators.

We now describe a subsampling estimator for the limit distribution of theorem
4.3 (Politis and Romano, 1994; Politis et al., 1999). Let $q>1,r\geq 0$, and
$l\geq 0$. Define $J(x)$ to be the limit distribution in eq. (12). Split the
set of units $\\{1,\dots,N\\}$ into all subsamples of size $b$ and index the
subsamples by $s$, $s=1,\dots,\binom{N}{b}$. Let $\vartheta^{(s)}_{b-k,b,T}$
be the $(b-k)$th order statistic in subsample $s$. Define the subsampling
estimator $L_{b,N,T}$ for $J$ as

$\displaystyle L_{b,N,T}(x)$
$\displaystyle=\dfrac{1}{\binom{N}{b}}\sum_{s=1}^{\binom{N}{b}}\operatorname{\mathbb{I}}\left\\{W_{s,b,N,T}\leq
x\right\\},\quad
W_{s,b,N,T}=\dfrac{\vartheta_{b-r,b,T}^{(s)}-\vartheta_{N-Nl/b,N,T}}{\vartheta_{b-q,b,T}^{(s)}-\vartheta_{b,b,T}^{(s)}},\quad\dfrac{Nl}{b}\leq
N.$ (15)

Observe that the subsample statistic $W_{s,b,N,T}$ is centered at
$\vartheta_{N-Nl/b,N,T}$. Intuitively, this corresponds to the $(1-l/b)$th
quantile, giving the correct centering to the subsampled statistics. If we are
interested in $F^{-1}(1)$, then $l=0$, and the statistic is centered at
$\vartheta_{N,N,T}$.

Define the estimated critical value $\hat{c}_{\alpha}$ as the $\alpha$th
quantile of $L_{b,N,T}$. The following result shows that $\hat{c}_{\alpha}$ is
consistent for the true critical values of interest for all $\alpha\in(0,1)$.

######  Theorem 4.4.

Let $b=N^{m},m\in(0,1)$. If $l>0$, let conditions of propositions 3.2 and 3.4
hold with $\delta=1-m$. If $\gamma<0$ and $l=0$, then let conditions of
proposition 3.2 hold. Then the subsampling estimator
$L_{b,N,T}(x)\xrightarrow{p}J(x)$ at all $x$ and
$\hat{c}_{\alpha}\xrightarrow{p}c_{\alpha}=J^{-1}(\alpha)$ for all
$\alpha\in(0,1)$.

######  Remark 7.

The assumptions and the proof of theorem 4.4 crucially depend on whether we
are interested in $F^{-1}(1)$ or in $F^{-1}(q)$ for $q<1$. If interest lies in
$F^{-1}(1)$, consistency of subsampling only requires checking whether the
extreme value theorem applies. If we are instead interested in $F^{-1}(q)$ for
$q<1$, the subsample statistics are centered at $\vartheta_{N-Nl/b,N,T}$, an
intermediate order statistic, in which case both the EVT and the IVT must
hold.

######  Remark 8.

Chernozhukov and Fernández-Val (2011) originally proposed and showed validity
of using subsampling for inference in the context of extreme quantile
regression. Theorem 4.4 shows that a similar approach can be applied in the
case of noisy observations. Further, it shows that subsampling is also
possible with $l=0$ in case of $\gamma<0$. The key challenge in establishing
the validity of subsampling is to control the estimation noise in the
subsamples and to leverage tail equivalence in the full sample.

######  Remark 9 (Estimation of the EV index).

Inference based on theorem 4.3 does not require an estimate of $\gamma$.
However, $\gamma$ may be consistently estimated. Let $k=k(N)$ satisfy
$k\to\infty,k=o(N)$. Let
$A_{N,T}=k^{-1}\sum_{i=0}^{k-1}\vartheta_{N-i,N,T}-\vartheta_{N-k,N,T}$ and
$B_{N,T}=k^{-2}\sum_{i=0}^{k-1}i\left(\vartheta_{N-i,N,T}-\vartheta_{N-k,N,T}\right)$.
The Hill (1975) estimator $\hat{\gamma}_{H}$ and probability weighted moment
(PWM) estimator $\hat{\gamma}_{PWM}$ of Hosking and Wallis (1987) are defined
as

$\displaystyle\hat{\gamma}_{H}$
$\displaystyle=\dfrac{1}{k}\sum_{i=0}^{k-1}\log(\vartheta_{N-i,N,T})-\log(\vartheta_{N-k,N,T}),\quad\hat{\gamma}_{PWM}=\dfrac{A_{N,T}-4B_{N,T}}{A_{N,T}-2B_{N,T}},$
(16)

If conditions of theorem 4.3 hold, then
$\hat{\gamma}_{H}\xrightarrow{p}\gamma$ if $\gamma\geq 0$ and
$\hat{\gamma}_{H}\xrightarrow{p}0$ if $\gamma<0$;
$\hat{\gamma}_{PWM}\xrightarrow{p}\gamma$ if $\gamma<1$ (approximately if $F$
has a finite first moment). The proof of consistency is given in proof
appendix.

$\hat{\gamma}_{H}$ and $\hat{\gamma}_{PWM}$ are likely to be imprecise if $N$
is not large. For the estimators to be close to $\gamma$, two conditions must
hold: $k$ must be sufficiently large and $k/N$ must be small. If $N$ is not
large, these requirements may be incompatible, as noted by Müller and Wang
(2017).555 We also note that estimating $\gamma$ provides an alternative way
of estimating the critical values necessary for application of theorem 4.3.
Simulation-based critical values can be obtained by drawing samples from the
right hand side distribution of eq. (12) after plugging in $\hat{\gamma}_{H}$
or $\hat{\gamma}_{PWM}$ in place of $\gamma$. However, subsampling-based
critical values of theorem 4.4 should be used unless $N$ is large.

######  Remark 10.

An alternative approach for constructing CIs for extreme quantiles using an
extreme order approximation is proposed by Müller and Wang (2017). They use
lemma 4.2 as the foundation of inference by treating the vector of top $q$
order statistics as a single draw from the corresponding limit distribution.
Based on such a joint EVT, Müller and Wang (2017) propose two methods:
inverting the likelihood ratio and minimizing the average expected length
where the average is taken over a pre-specified range of values for $\gamma$.
Both methods lead to valid CIs for extreme quantiles in our setting, as long
as lemma 4.2 holds; though we recommend using CIs based on theorem 4.3 that
require no optimization and no bounds on the parameter $\gamma$.

####  4.2 Inference Using Intermediate Order Approximations

The intermediate order approximation of theorem 3.3 provides an alternative
approach to inference that is based on convergence of intermediate order
statistics (eq. (4)). In this case, the quantile of interest is modeled as
drifting to 1 at a rate $k/N$ where $k\to\infty,k=o(N)$ as $N\to\infty$; this
rate is slower than the rate $N^{-1}$ of extreme order approximations. The
resulting statistic is asymptotically standard normal. Intermediate order
approximations should not be applied to $F^{-1}(\tau)$ for $\tau$ very close
to 1 (see remark 11 below).

As the scaling rate $c_{N}$ of statistic (4) is unknown, we follow an approach
similar to the feasible EVT 4.3 and use a second intermediate order statistics
to eliminate $c_{N}$.666 There are alternative approaches to inference using
intermediate order statistics. For example, the subsampling approach of
Bertail et al. (1999, 2004) can be used to account for unknown scaling rate
$c_{N}$. However, the presence of a slowly varying component in $c_{N}$
requires using multiple subsampling sizes, which may be problematic if $N$ is
not extremely large. Alternatively, for example, see ch. 4 of de Haan and
Ferreira (2006) for inference under a second-order condition in a setting
without estimation noise.  Unlike the case of theorem 4.3, the statistics in
the denominator of the self-normalized statistic are asymptotically perfectly
dependent and only differ by a non-zero deterministic factor. The following
theorem first establishes that such a technique works in the noiseless case,
which may be of independent interest; the result is then transferred to noisy
observables.

######  Theorem 4.5.

Let assumptions 1 and 4 hold. Let $k=o(N),k\to\infty$. Let $f=F^{\prime}$ be
non-increasing or non-decreasing in some left neighborhood of $F^{-1}(1)$
$(F^{-1}(1)\leq\infty)$.

  1. (1)

Then

$\dfrac{\theta_{N-k,N}-F^{-1}\left(1-\frac{k}{N}\right)}{\theta_{N-k,N}-\theta_{N-k-\lfloor\sqrt{k}\rfloor,N}}\Rightarrow
N(0,1),\quad N\to\infty.$ (17)

  2. (2)

In addition, let conditions of theorem 3.3 hold when evaluated at $k$ and
$k+\sqrt{k}$. Then

$\dfrac{\vartheta_{N-k,N,T}-F^{-1}(1-k/N)}{\vartheta_{N-k,N,T}-\vartheta_{N-k-\lfloor\sqrt{k}\rfloor,N,T}}\Rightarrow
N(0,1),\quad N,T\to\infty.$ (18)

The choice of $k$ determines the centering quantile of interest. When $k$ is
chosen, the denominator is uniquely determined by $k$. This is in contrast to
the free choice of the parameters $r,q$ in the statistics of theorem 4.3.

######  Example 6.

Let $N=200$ and suppose that $k=k(N)$ is such that $k(200)=16$ and
$k\to\infty,k=o(N)$ as $N\to\infty$. Then by theorem 4.5 a confidence interval
for $F^{-1}(0.92)$ based on sample of size $N=200$

$\left[\vartheta_{N-k,N}-z_{1-\alpha/2}\left(\vartheta_{N-k,N}-\vartheta_{N-k-\lfloor\sqrt{k}\rfloor,N}\right),\vartheta_{N-k,N}-z_{\alpha/2}\left(\vartheta_{N-k,N}-\vartheta_{N-k-\lfloor\sqrt{k}\rfloor,N}\right)\right],$

where $z_{\alpha}$ the $\alpha$th quantile of the standard normal
distribution. As with extreme approximations (example 3), the target quantity
of the interval $F^{-1}(1-k/N)$, though the quantile is drifting to 1 at a
rate $k/N$, which is slower than $1/N$.

######  Remark 11.

The approximation of theorem 4.5 should not be used if $k$ is small. If
$\lfloor\sqrt{k}\rfloor$ is small but positive, $\vartheta_{N-k,N}$ and
$\vartheta_{N-k-\lfloor\sqrt{k}\rfloor,N}$ will be close, and the statistic
will be unstable. If $\lfloor\sqrt{k}\rfloor=0$, the statistic cannot be used
at all.

####  4.3 Inference Using Central Order Approximations

The third method of inference is based on the central limit theorem for
quantiles. The quantile of interest $F^{-1}(\tau)$ is modeled as fixed and
independent from sample size $(N,T)$, in contrast to the extreme and
intermediate approximations given above. Such “central” order approximations
require that a sufficient number of observations be available on both sides of
the corresponding sample order statistic $\vartheta_{\lfloor
N\tau\rfloor,N,T}$.

Jochmans and Weidner (2022) derive such approximations in the context of our
problem, and we briefly state their results. They study a version of (1) given
by $\vartheta_{i,T}=\theta_{i}+T^{-1/2}\varepsilon_{i}$ (that is, $p=1/2$ and
$G_{T}=G$ for all $T$). We introduce some additional notation: let $K$ be a
kernel function, $h$ a bandwidth parameter, and define

$\displaystyle b_{F}(x)$
$\displaystyle=\left[\dfrac{\operatorname{\mathbb{E}}\left[\sigma^{2}_{i}|\theta_{i}=x\right]f(t)}{2}\right]^{\prime},\quad\sigma_{i}^{2}=\operatorname{\mathrm{Var}}(\varepsilon_{i}|\theta_{i}),$
(19) $\displaystyle\hat{b}_{F}$
$\displaystyle=-\dfrac{(nh^{2})^{-1}\sum_{i=1}^{n}\sigma^{2}_{i}K^{\prime}((\vartheta_{i,T}-\theta)/h)}{2},\quad\hat{\tau}^{*}=\tau+\dfrac{\hat{b}_{F}(\vartheta_{\lfloor
N\tau\rfloor,N,T})}{T}.$ (20)

$\sigma_{i}^{2}$ is assumed known and invariant over time.

######  Theorem 4.6 (Propositions 2 and 4 of Jochmans and Weidner (2022)).

Let conditions of proposition 3 in Jochmans and Weidner (2022) hold, and in
particular let for all $T$ $\varepsilon_{i,T}=\varepsilon_{i}$,
$\operatorname{\mathbb{E}}[\varepsilon_{i}|\theta_{i}]=0$, and
$\varepsilon_{i}$ be independent from $\theta_{i}$ given $\sigma_{i}^{2}$. Let
$\tau\in(0,1)$.

  1. (1)

If $N/T^{2}\to c<\infty$, then as $N,T\to\infty$

$\sqrt{N}\left(\vartheta_{\lfloor
N\tau\rfloor,N,T}-F^{-1}(\tau)+\dfrac{1}{T}\dfrac{b_{F}(F^{-1}(\tau))}{f(F^{-1}(\tau))}\right)\Rightarrow
N\left(0,\dfrac{\tau(1-\tau)}{f(F^{-1}(\tau))^{2}}\right).$ (21)

  2. (2)

If $N/T^{4}\to 0$, then as $N,T\to\infty$

$\sqrt{N}\left(\vartheta_{\lfloor
N\hat{\tau}^{*}\rfloor,N,T}-F^{-1}(\tau)\right)\Rightarrow
N\left(0,\dfrac{\tau(1-\tau)}{f(F^{-1}(\tau))^{2}}\right).$ (22)

Theorem 4.6 is a restatement of proposition 2 and 4 of Jochmans and Weidner
(2022). It shows that the sample $\tau$th quantile $\vartheta_{\lfloor
N\tau\rfloor,N,T}$ is a consistent and asymptotically normal estimator for
$F^{-1}(\tau)$ with standard asymptotic variance. However, $\vartheta_{\lfloor
N\tau\rfloor,N}$ is subject to bias of leading order $1/T$. Jochmans and
Weidner (2022) show that this bias can be reduced by instead considering the
sample $\hat{\tau}^{*}$th quantile: $\vartheta_{\lfloor
N\hat{\tau}^{*}\rfloor,N,T}$ is consistent and asymptotically normal with the
same variance, but the leading order of the bias is instead given by
$1/T^{2}$. This bias is eliminated if $\sqrt{N}/T^{2}\to 0$.

######  Remark 12.

For central order approximations, the order of the bias incurred by using
$\vartheta_{\lfloor N\tau\rfloor,N,T}$ in place of $\theta_{\lfloor
N\tau\rfloor,N}$ is the same for a broad class of distributions, and equal to
1/T. This invariance of bias order enables construction of the debiased
estimator $\vartheta_{\lfloor N\hat{\tau}^{*}\rfloor,N,T}$. The situation is
more complex for extreme and intermediate order approximations. The magnitude
of the impact of estimation noise is determined by the interaction of $a_{N}$
and $T$ in theorem 3.1; $a_{N}$ itself may behave like $N^{\gamma}$ for
$\gamma\in\operatorname{\mathbb{R}}$ depending on $F$, up to slowly varying
components.

###  5 Simulation Study

We assess the performance of our confidence intervals with a simulation study.
We consider a linear regression model with unit-specific coefficients where
the outcome $y_{it}$ is generated by covariates $(x_{it},z_{it})$ as

$y_{it}=\alpha_{i}+\eta_{i}x_{it}+\theta_{i}z_{it}+\sqrt{\dfrac{\operatorname{\mathrm{Var}}(\theta_{i})}{\operatorname{\mathrm{Var}}(u_{it})}}\times
u_{it}.$ (23)

The parameter of interest $\theta_{i}$ is the coefficient on $z_{it}$. We are
interested in the coverage and length properties of a nominal 95% confidence
interval for 0.85-0.999th quantiles.

The data is generated as follows. Define
${F}_{Fr,\kappa}(\theta)=1-(\theta+1)^{-\kappa},\kappa>0,\theta\in[0,\infty)$.
$F_{Fr,\kappa}$ is a canonical example of a one-sided power law distribution
whose moments of order below $\kappa$ are finite. $F_{Fr,\kappa}$ satisfies
assumption 2; its EV index $\gamma$ is $1/\kappa>0$. Coefficients
$(\alpha_{i},\eta_{i},\theta_{i})$ are drawn from a trivariate Gaussian copula
with correlation 0.3 and the same marginals $F_{Fr,\kappa}$ for $\kappa=4$.
This choice of $\kappa$ matches the data of our empirical application.
Covariates $x_{it}$ are generated as
$0.3\eta_{i}+\sigma_{x}\left(1+0.3\left\lVert(\alpha_{i},\eta_{i},\theta_{i})\right\rVert\right)^{1/2}(0.1+U_{i})$
where $U_{i}$ is a Uniform[0, 1]; $z_{it}$ are generated similarly with
$\theta_{i}$ in place of $\eta_{i}$. This is a stylized setup in which UIH and
covariates are positively dependent. $u_{it}$ is sampled independently from
the coefficients and the covariates from $G_{\beta}$, where $G_{\beta}$ has
density $g_{\beta}=\beta(1+\lvert x\rvert)^{-\beta-1}/2$ for
$x\in\operatorname{\mathbb{R}}$ and $\beta=8$. $G_{\beta}$ is a two-sided
power law with finite moments of order $<\beta$. $u_{it}$ is rescaled so that
its variance matches that of $\theta_{i}$. Coefficients are estimated using
OLS. As a result, the estimation noise $\varepsilon_{i,T}$ is not independent
from $\theta_{i}$. We consider two cross-sectional sizes: $N=200$ and
$N=1000$. In both cases $T=10$, which matches the time dimension of the
dataset of our empirical application. All results are based on 10000 Monte
Carlo samples.

We construct the confidence intervals using the three approximations given in
section 4.

  1. (1)

_Extreme_ : intervals based on the the feasible EVT 4.3. Quantiles of the
limit distribution are estimated using subsampling as in theorem 4.4. We
choose $r$ in line with remark 6. The “mixed” CI is centered at the
corresponding sample quantile for each quantile of interest, that is,
$r=\lfloor l\rfloor$ (line 1a on figs. 1 and 2). The “max only” CI has $r=0$
and is centered at the maximum for all quantiles (line 1b). In both cases we
pick $q=2$ (unreported simulation evidence shows that performance of the
intervals does not depend on choice of $q$ in the range 2-10). Additionally,
for $N=2000$ we consider the “mixed” CI with critical values obtained by
simulating from the limit distribution after plugging in a consistent
estimator of $\gamma$ (line 1c). We use the PWM estimator of remark 9. For
$N=200$, $\gamma$ is not estimated reliably, and we only report the results of
this method in the online appendix, along with the similar results of using
the Hill estimator.

  2. (2)

_Intermediate_ : we construct to CIs based on the statistic of theorem 4.5
that differ in how the critical values are estimated: using the limiting
standard normal distribution (line 2a) and by subsampling (line 2b).

For comparison, we also include a textbook confidence interval based on the
extrapolation estimator of theorem 4.3.1 in de Haan and Ferreira (2006) (line
2c). In line with their simulation evidence, we pick
$k=\lfloor\sqrt{N}\rfloor$ and use the PWM estimator of remark 9. This CI can
only be constructed for sufficiently high quantiles. The validity of the
extrapolation interval hinges on a second-order condition (de Haan and
Resnick, 1996) which we do not examine in the current paper.

  3. (3)

_Central_ : we reports two CIs. The first interval is based on the raw data
(line 3a). We construct the CI using the raw $\vartheta_{i,T}$ using a
binomial CI. The same approach is implemented in the Stata command centile.
The second interval uses the analytical correction of Jochmans and Weidner
(2022) (line 3b), based on theorem 4.6. As advocated by Jochmans and Weidner
(2022), we report bootstrap confidence intervals with 1000 bootstrap samples.

Figure 1: Coverage and length for 95% nominal confidence interval.
$F=F_{Fr,\kappa},\kappa=4$ (power law tail). $u_{it}\sim G_{\beta},\beta=8$.
$T=10$. Top panel: $N=200$. Bottom panel: $N=2000$.

We briefly discuss the validity of different approximations. The rate
conditions of theorem 4.3 hold for both sample sizes by proposition 3.2 as the
estimation noise has more moments than $\theta_{i}$. The sufficient conditions
for theorem 4.5 hold only for a range of higher quantiles due to the
relationship between $k$ and the quantile targeted in theorem 4.5, see remark
5. For central order approximations, the rate conditions for validity of using
raw data do not hold, especially in the second sample, see theorem 4.6(1). The
rate conditions for the analytical correction of Jochmans and Weidner (2022)
hold in both samples.

We present our results on figure 1. Fig. 1 depicts coverage rates and lengths
for the above confidence intervals for the two values of $N$ considered. In
order to assess the impact of estimation noise, in fig. 2 we plot the same
results in a noiseless setting, that is, with $u_{it}=0$.

For inference on extreme quantiles, we recommend using extreme approximations.
For smaller cross-section sample sizes, the quantiles of the limiting
distribution should be estimated by subsampling. For larger samples, both
subsampling and simulating from the limit distributions are viable options.
Based on length properties, the mixed approach (1a and 1c) displays the best
overall performance, and we recommend using it.

The properties of extreme approximations worsen somewhat further away from the
tail, when there are many points available to the right of the corresponding
sample quantile. The max-based and the mixed CIs are affected differently.
First, consider the mixed CI (1a and 1c). Its coverage deteriorates somewhat
for quantiles such that there are more than 100 observations to the right of
the corresponding sample quantile, that is, below the 0.95th quantile on the
second panel of figs. 1 and 2. This effect is primarily driven by estimation
noise. To see this, note that the behavior of $\vartheta_{N-r,N,T}$ in the
numerator of eq. (12) changes when $r$ is relatively large:
$\vartheta_{N-r,N,T}$ behaves more similarly to an intermediate statistic.
However, the rate conditions of the IVT do not hold in that section of the
tail, and estimation noise distorts inference (compare the second panels of
figs. 1 and 2). The problem is alleviated by using simulated critical values
(1c), effectively imposing the EV limit distribution. Second, the max-based CI
(1b) only uses the top statistics and its validity is not affected. However,
it has unfavorable length properties for quantiles that are sufficiently far
from the tail.

Figure 2: Coverage and length for 95% nominal confidence interval.
$F=F_{Fr,\kappa},\kappa=4$ (power law tail). Noiseless data. $T=10$. Top
panel: $N=200$. Bottom panel: $N=2000$.

Central approximations suffer from the two problems highlighted in the
introduction: effect of estimation noise and failure of central approximations
in the tail. The first issue is evident in the fairly poor performance of CIs
based on raw data for quantiles below 0.99, especially for $N=2000$ (3a). The
analytical correction of Jochmans and Weidner (2022) uniformly improves
coverage for all such quantiles (3b). The second issue is common to both using
raw data and using the correction; coverage and length of confidence intervals
based on central approximations collapse to 0 for high quantiles, an effect
most notable for $N=200$.

Confidence intervals based on the intermediate approximation of theorem 4.5
are a viable option in the regions where theorem 4.5 holds and the statistic
(18) is stable (2a and 2b). This corresponds to 0.95-0.99th quantiles for
$N=200$ and 0.95-0.997th quantiles for $N=2000$. We recommend subsampling (2b)
to estimate critical values, as the normal distribution does not provide a
good approximation for the values of $N$ considered. Theorem 4.5 should not be
used if $\lfloor\sqrt{k}\rfloor$ is small, as remark 11 notes. In this case
the statistic (18) is unstable. This instability is correctly picked up by the
subsampling procedure, resulting in very wide confidence intervals. This
affects quantiles beyond the 0.99th for $N=200$ and the 0.997th for $N=2000$.
Finally, the coverage properties of extrapolation-based CI (2c) stem from its
extreme length. Extrapolation-based CIs are longer that all other
approximations barring that of theorem 4.5 in the unstable region. This growth
of interval length is sufficient to offset the impact of estimation noise.

We note that in the online appendix we provide additional detailed simulations
for distributions $F$ which satisfy assumption 2 with $\gamma=0$ and
$\gamma<0$, along with results for $u_{it}\sim N(0,1)$. Furthermore, we also
explore performance of the corrected estimators formed as in eq. (14).

###  6 Empirical Application

As an empirical illustration, we revisit the question of differences in the
distribution of firm productivity between denser and less dense areas in the
setting of Combes et al. (2012) (CDGPR12). As firm productivity has to be
estimated from firm-level data, this provides a natural example of the setting
of this paper.

##### Background

We first provide a brief summary of the relevant aspects of CDGPR12. It is
well-established that firms tend to be more productive in more densely
populated areas (Melo et al., 2009). CDGPR12 seek to disentangle the two
principal possible sources of differences – agglomeration economies and firm
selection. To do so, CDGPR12 propose a two-step procedure. In the first step
they estimate the unobserved firm-specific productivities. In the second step
they compare the distribution of productivities in areas with above-median
employment density (AAMED) with the distribution in areas with below-median
employment density (ABMED) using the first-step estimates. In the second stage
CDGPR12 make the key assumption that the distributions of true unobserved
productivity in AAMED and ABMED are formed from the same latent parent
distribution and only differ in three parameters – mean, variance, and
proportion of the left tail truncated. This assumption generates a continuum
of moment conditions which CDGPR12 then exploit to estimate the differences in
each parameter between AAMED and ABMED. The magnitudes of the differences in
parameters reflect the impact of agglomeration economies and firm selection in
AAMED vis-à-vis ABMED. The mean and variance reflect the effects of
agglomeration economies, which make firms more productive on average; however,
some firms are likely to benefit more than others. The truncation parameter
reflects the effects of firm selection. As Asplund and Nocke (2006) argue,
competition is tougher in larger markets (such as larger cities). In the
setting of CDGPR12, this translates into stronger truncation of the left tail
of the latent parent distribution; the minimal level of productivity required
to survive in a larger market is higher than in a smaller one (up to mean and
variance).

##### Empirical questions

In this application we consider two questions. First, the key question of
interest is whether the aforementioned three parameters are sufficient to
capture the difference between the tails of AAMED and the ABMED distributions.
A finding that the tails are identical up to these parameters would support
the key assumption of the second estimation step of CDGPR12. Second, we
directly compare the left and the right tails of the AAMED and the ABMED
distributions with no adjustments. Comparing the right tails reveals the
magnitude of the productivity advantage enjoyed by the most productive firms
of AAMED relative to the most productive firms of ABMED. Analyzing the left
tail allows us to examine the evidence for firm selection. The approach
developed by CDGPR12 is only capable of estimating the difference in the
proportion of the left tail that is truncated. It does not reveal whether
truncation occurs at all, and at what level.

##### Data

We use the publicly available replication data provided by
CDGPR12.777Available from D. Puga’s website at
https://diegopuga.org/data/selectagg/ The dataset includes firm-level
productivity estimates, labeled as belonging either to AAMED. From each
original dataset we randomly select $N=2000$ firms, in line with our
simulation study (the full cross-section contains $58654$ firms for AAMED and
$78359$ firms for ABMED). Estimates are constructed by CDGPR12 using data
covering 1994-2002, and thus $T=9$. We give a brief description of the
estimation procedure for productivity, and refer to CDGPR12 for a detailed
discussion of the data and the estimation procedure employed. Firm $i$ in
sector $s$ generates value added in period $t$ according to a Cobb-Douglas
function of the form

$V_{i\,t}=\exp(\theta_{i})K_{i\,t}^{\beta_{1,s}}L_{i\,t}^{\beta_{2,s}}\exp(u_{i\,t}+\beta_{0,t,s})$
(24)

where $\theta_{i}$ is the time-invariant log total factor productivity (TFP)
of firm $i$, $V_{i\,t}$ is value added, $K_{i\,t}$ is capital, $L_{i\,t}$ is a
labor index, $u_{i\,t}$ is a mean-zero term that combines individual
measurement error and time-varying productivity shocks,
$\beta_{1,s},\beta_{2,s}$ are sector-specific factor shares, and
$\beta_{0,t,s}$ is a sector- and time-specific intercept. $\theta_{i}$ is the
key measure of productivity. Firms in AAMED draw $\theta_{i}$ from some latent
distribution $F_{AM}$, firms in ABMED draw $\theta_{i}$ from $F_{BM}$.888In
principle, the distributions may be sector-specific. However, the data of
CDGPR12 does not include sectoral labels, and we thus focus on the results
across all sectors (compare the last line in table 1 of CDGPR12). The noisy
estimates $\vartheta_{i,T}$ are constructed in two steps. First, sector-
specific production functions are estimated. Second, firm-specific log TFP is
estimated as the average residual from the estimated production function, that
is, $\vartheta_{i,T}=T^{-1}\sum_{t=1}^{T}[\log
V_{i\,t}-\hat{\beta}_{0,s,t}-\hat{\beta}_{1,s}\log
K_{i\,t}-\hat{\beta}_{2,s}\log L_{i\,t}]$. Last, the data is normalized so
that average log TFP across ABMED is zero.

##### Connection to the literature on estimation of the production frontier

We highlight that our analysis (and that of CDGPR12) follows the literature on
production function estimation; see Ackerberg et al. (2007) for a review. This
should be contrasted with the literature on production frontier estimation
(e.g. Kneip et al. (2015); Kumbhakar et al. (2020a, b)). In our setting, each
firm may be viewed as always being on its production frontier. Each frontier
is characterized by eq. (24), and interest centers on these firm-specific
parameters, rather than an economy- or sector-wide production frontier.

##### Assumptions and rate conditions

We make two key assumptions in our analysis, one related to the estimation
error and the other to the underlying distributions $F_{AM}$ and $F_{BM}$.
First, we assume that the estimation noise affecting $\vartheta_{i,T}$ has at
least 8 moments. The principal source of error in $\vartheta_{i,T}$ are the
measurement errors in the value-added and year- and firm-specific productivity
shocks; the estimation noise in $(\beta_{0,s,t},\beta_{1,s},\beta_{2,s})$ is
negligible due to the large size of the cross-section. Correspondingly, this
assumptions only concerns measurement error and deviations from average
productivity. Second, we assume that both the left and right tails of $F_{AM}$
and $F_{BM}$ are infinite. This is well-established in the literature for the
right tail of the productivity distribution (Melo et al., 2009). For the left
tail the assumption can be motivated as follows. The dataset covers
(approximately) all firms that exist in the estimation period with no
restriction on firm age. We therefore interpret $F_{AM}$ and $F_{BM}$ as
stationary distributions that account for firm entry and exit, in line with
interpretation of Asplund and Nocke (2006). Correspondingly, $F_{AM}$ and
$F_{BM}$ also include unproductive firms that have not yet exited the market.
We note that this assumption is supported by the data, as we show below.

Under the above assumptions the rate conditions of theorem 4.3 hold in line
with proposition 3.2 and our simulation evidence. A sufficient rate condition
is $N/T^{4}\to 0$, which we assume to be broadly satisfied for the sample size
at hand.999Additionally, under the same rate condition Jochmans and Weidner
(2022) show in their simulation study that noisy estimates are informative for
panels of size $N=200,T=5$. We observe that $200/5^{4}\approx 2000/9^{4}$.

##### EV index estimation and evaluation of rate conditions

We begin by estimating the EV indices $\gamma$ for the left and the right
tails of the AAMED and ABMED distributions. The estimates of $\gamma$ for
$F_{BM}$ are 0.36 for the left tail and 0.32 for the right tail. The estimates
for $F_{AM}$ are 0.35 for the left tail and 0.30 for the right tail. These
estimates of $\gamma$ are computed as the average of the two estimators of
remark 9 for $k=\lfloor N^{3/5}\rfloor$; in the online appendix we show that
our estimates are robust to choice of estimator and choice of $k$. The
estimates for $\gamma$ are large and compatible with fairly heavy-tailed
distributions. The estimates for both tails of the $F_{BM}$ and $F_{AM}$ are
compatible with a distribution with three but not four finite moments.

There are two principal observations that we may make based on our estimates
of $\gamma$. First, the heavy-tailed behavior serves as a confirmation of the
assumption that $F_{AM}$ and $F_{BM}$ have infinite tails. Recall that we
assume that the measurement error and deviations from $\theta_{i}$ have at
least eight moments. Under this assumption, the EV index of the estimation
noise distribution cannot exceed 0.125. Consequently, we would expect to see
lower estimates of $\gamma$ if the tail of the distribution of estimation
noise dominated the tails of $F_{AM}$ and $F_{BM}$. Second, in line with our
interpretation of $F_{AM}$ and $F_{BM}$, the data does not support the
existence of a sharp minimal threshold for productivity required for survival,
at least in the sense of a barrier to entry. A threshold for survival may
exist but only apply to a given firm in the long run.

##### Comparison of standardized tails

We now examine whether the difference between the tails of $F_{AM}$ and
$F_{BM}$ can be captured by the three parameters of CDGPR12. In order to do
so, we first standardize both the AAMED and the ABMED datasets to have the
same mean and variance. We note that the positive estimates of the EV index
for left tails of $F_{AM}$ and $F_{BM}$ are incompatible with truncation.
Correspondingly, we do not modify the tails of the data. Using adjusted data,
we compute 95% confidence intervals for the tail quantiles using theorem 4.3.
We take $r=\lfloor l\rfloor$ and $q=10$ and estimate critical values by
subsampling. The quantiles considered are 0.001-0.1 for the left tail and
0.9-0.999 for the right tail. We note that the results are robust to choice of
method of confidence interval. In the online appendix we report intervals
based on all the methods considered in our simulation study and find that the
results are virtually identical regardless of the method used.

Our key finding is that the tails of the AAMED and the ABMED distribution are
similar up to mean and variance of $(F_{AM},F_{BM})$. Both tails are
statistically identical, as fig. 3 shows. Overall, this finding supports the
key assumption CDGPR12 make in their second estimation stage.

Figure 3: 95% confidence intervals for extreme quantiles (between
corresponding lines), AAMED and ABMED data standardized to have the same mean
and variance. CIs based on theorem 4.3 with critical values estimated by
subsampling

Figure 4: 95% confidence intervals for extreme quantiles of AAMED and ABMED
distributions (between corresponding lines). CIs based on theorem 4.3 with
critical values estimated by subsampling

##### Comparison of non-standardized tails

We now turn to comparing the left and the right tails of $F_{AM}$ and $F_{BM}$
with no standardization applied to the data. We construct 95% confidence
intervals for the 0.001-0.1st and 0.9-0.999th quantiles using theorem 4.3 as
above, with the results depicted on fig. 4. We first consider the left tails.
As noted above, positive estimates of $\gamma$ suggest that
$F^{-1}_{AM}(0)=F^{-1}_{BM}(0)=-\infty$. Moreover, as the top panel of fig. 4
indicates, there is no difference between the left tails of $F_{AM}$ and
$F_{BM}$. In contrast, the right tails of $F_{AM}$ and $F_{BM}$ are
statistically different up to the 0.995th quantile, showing an overall
productivity advantage of AAMED relative to ABMED. However, this result should
be interpreted carefully in light of differences in sectoral composition
between AAMED and ABMED. Sectoral labeling would necessary to quantify the
exact magnitude of the productivity advantage. As the publicly available data
from Combes et al. (2012) does not include this labeling, we leave a detailed
investigation of the point to future research.

##### Additional results

We remark that we provide additional robustness checks in the online appendix,
including detailed plots for all the methods used in this paper, further
details on estimation of $\gamma$, and results obtained using the full
dataset.

### References

  * Ackerberg et al. (2007) D. A. Ackerberg, C. L. Benkard, S. Berry, and A. Pakes.  Econometric Tools for Analyzing Market Outcomes.  In J. Heckman and E. Leamer, editors, _Handbook of Econometrics, vol. 6A_ , chapter 63, pages 4171–4276. North-Holland, Amsterdam, 2007.
  * Arellano and Bonhomme (2012) M. Arellano and S. Bonhomme.  Identifying distributional characteristics in random coefficients panel data models.  _Review of Economic Studies_ , 79(3):987–1020, 2012.  ISSN 00346527.  doi: 10.1093/restud/rdr045.
  * Asplund and Nocke (2006) M. Asplund and V. Nocke.  Firm Turnover in Imperfectly Competitive Markets.  _Review of Economic Studies_ , 73(2):295–327, 2006.  ISSN 00346527.  doi: 10.1111/j.1467-937X.2006.00377.x.
  * Barras et al. (2021) L. Barras, P. Gagliardini, and O. Scaillet.  Skill, Scale and Value Creation in the Mutual Fund Industry.  2021\.  doi: 10.2139/ssrn.3269995.
  * Bertail et al. (1999) P. Bertail, D. N. Politis, and J. P. Romano.  On Subsampling Estimators with Unknown Rate of Convergence.  _Journal of the American Statistical Association_ , 94(446):569–579, 1999.  ISSN 1537274X.  doi: 10.1080/01621459.1999.10474151.
  * Bertail et al. (2004) P. Bertail, C. Haefke, D. N. Politis, and H. White.  Subsampling The Distribution of Diverging Statistics with Applications to Finance.  _Journal of Econometrics_ , 120(2):295–326, 2004\.  ISSN 03044076.  doi: 10.1016/S0304-4076(03)00215-X.
  * Bingham et al. (1987) N. H. Bingham, C. M. Goldie, and J. L. Teugels.  _Regular Variation_.  Cambridge University Press, Cambridge, 1987.  ISBN 9780521379434.  doi: 10.1017/CBO9780511721434.
  * Browning and Carro (2007) M. Browning and J. M. Carro.  Heterogeneity and Microeconometrics Modeling.  _Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress, Volume III_ , pages 47–74, 2007.  doi: 10.1017/CBO9780511607547.004.
  * Browning and Carro (2010) M. Browning and J. M. Carro.  Heterogeneity in dynamic discrete choice models.  _The Econometrics Journal_ , 13(1):1–39, 2010\.  ISSN 13684221.  doi: 10.1111/j.1368-423X.2009.00301.x.
  * Brownlees and Morozov (2022) C. Brownlees and V. Morozov.  Unit Averaging for Heterogeneous Panels.  2022\.
  * Chernozhukov (2005) V. Chernozhukov.  Extremal Quantile Regression.  _Annals of Statistics_ , 33(2):806–839, 2005\.  ISSN 00905364.  doi: 10.1214/009053604000001165.
  * Chernozhukov and Fernández-Val (2011) V. Chernozhukov and I. Fernández-Val.  Inference for Extremal Conditional Quantile Models, With An Application to Market and Birthweight Risks.  _Review of Economic Studies_ , 78(2):559–589, 2011.  ISSN 00346527.  doi: 10.1093/restud/rdq020.
  * Combes et al. (2012) P. P. Combes, G. Duranton, L. Gobillon, D. Puga, and S. Roux.  The Productivity Advantages of Large Cities: Distinguishing Agglomeration From Firm Selection.  _Econometrica_ , 80(6):2543–2594, 2012.  ISSN 0012-9682.  doi: 10.3982/ecta8442.
  * de Haan and Ferreira (2006) L. de Haan and A. Ferreira.  _Extreme Value Theory_.  Springer, 2006.  ISBN 978-0-387-23946-0.  doi: 10.1007/0-387-34471-3.
  * de Haan and Resnick (1996) L. de Haan and S. I. Resnick.  Second-Order Regular Variation and Rates of Convergence in Extreme-Value Theory.  _The Annals of Probability_ , 24(1):97–124, 1996\.
  * Dekkers and de Haan (1989) A. L. Dekkers and L. de Haan.  On the Estimation of the Extreme-Value Index and Large Quantile Estimation.  _The Annals of Statistics_ , 17(4):1795–1832, 1989.  doi: 10.1214/aos/1176347396.
  * Evdokimov (2010) K. Evdokimov.  Identification and Estimation of a Nonparametric Panel Data Model with Unobserved Heterogeneity.  2010\.
  * Gabaix (2009) X. Gabaix.  Power Laws in Economics and Finance.  _Annual Review of Economics_ , 1:255–293, 2009.
  * Gabaix (2016) X. Gabaix.  Power Laws in Economics: An Introduction.  _Journal of Economic Perspectives_ , 30(1):185–205, 2016.
  * Girard et al. (2021) S. Girard, G. Stupfler, and A. Usseglio-Carleve.  Extreme Conditional Expectile Estimation In Heavy-Tailed Heteroscedastic Regression Models.  _Annals of Statistics_ , 49(6):3358–3382, 2021\.  ISSN 21688966.  doi: 10.1214/21-AOS2087.
  * Gnedenko (1943) B. V. Gnedenko.  Sur La Distribution Limite Du Terme Maximum D’Une Serie Aleatoire.  _Annals of Mathematics_ , 44(3):423–453, 1943\.  ISSN 0003486X.  doi: 10.2307/1968974.
  * Heckman (2001) J. J. Heckman.  Micro Data, Heterogeneity and the Evaluation of Public Policy: Nobel Lecture.  _Journal of Political Economy_ , 109(4):673–748, 2001.  ISSN 23281235.  doi: 10.1086/322086.
  * Higgins et al. (2009) J. P. Higgins, S. G. Thompson, and D. J. Spiegelhalter.  A Re-Evaluation of Random-Effects Meta-Analysis.  _Journal of the Royal Statistical Society: Series A_ , 172(1):137–159, 2009.  doi: 10.1111/j.1467-985x.2008.00552.x.
  * Hill (1975) B. M. Hill.  A Simple General Approach to Inference About the Tail of a Distribution.  _The Annals of Statistics_ , 3(5):1163–1174, 1975\.
  * Hosking and Wallis (1987) J. R. M. Hosking and J. R. Wallis.  Parameter and Quantile Estimation for the Generalized Pareto Distribution.  _Technometrics_ , 29(3):339–349, 1987.  doi: 10.2307/1269343.
  * Jochmans and Weidner (2022) K. Jochmans and M. Weidner.  Inference on a Distribution From Noisy Draws.  _Econometric Theory_ , pages 1–38, 2022.  doi: 10.1017/S0266466622000378.
  * Kneip et al. (2015) A. Kneip, L. Simar, and I. Van Keilegom.  Frontier Estimation in the Presence of Measurement error with Unknown Variance.  _Journal of Econometrics_ , 184(2):379–393, 2015\.  ISSN 18726895.  doi: 10.1016/j.jeconom.2014.09.012.  URL http://dx.doi.org/10.1016/j.jeconom.2014.09.012.
  * Kumbhakar et al. (2020a) S. C. Kumbhakar, C. F. Parmeter, and V. Zelenyuk.  Stochastic Frontier Analysis: Foundations and Advances I.  In _Handbook of Production Economics_ , pages 1–40. 2020a.  doi: 10.1007/978-981-10-3450-3_9-2.
  * Kumbhakar et al. (2020b) S. C. Kumbhakar, C. F. Parmeter, and V. Zelenyuk.  Stochastic Frontier Analysis: Foundations and Advances II.  In _Handbook of Production Economics_ , pages 1–38. 2020b.  doi: 10.1007/978-981-10-3450-3_11-1.
  * Makarov (1981) G. Makarov.  Estimates for the Distribution Function of a Sum of Two Random Variables When the Marginal Distributions are Fixed.  _Theory of Probability Amd Its Applications_ , 26(4):803–806, 1981.  doi: 10.1137/1126086.
  * Melo et al. (2009) P. C. Melo, D. J. Graham, and R. B. Noland.  A Meta-Analysis of Estimates of Urban Agglomeration Economies.  _Regional Science and Urban Economics_ , 39(3):332–342, 2009.  ISSN 01660462.  doi: 10.1016/j.regsciurbeco.2008.12.002.
  * Müller and Wang (2017) U. K. Müller and Y. Wang.  Fixed-k Asymptotic Inference About Tail Properties.  _Journal of the American Statistical Association_ , 112(519):1334–1343, 2017.  ISSN 1537274X.  doi: 10.1080/01621459.2016.1215990.
  * Nelson and Startz (1990) C. R. Nelson and R. Startz.  Some Further Results on the Exact Small Sample Properties of the Instrumental Variable Estimator.  _Econometrica_ , 58(4):967–976, 1990.  doi: 10.2307/2938359.
  * Okui and Yanagi (2019) R. Okui and T. Yanagi.  Kernel estimation for panel data with heterogeneous dynamics.  _The Econometrics Journal_ , 23(1):156–175, 2019\.  ISSN 1937240X.  doi: 10.1093/ectj/utz019.
  * Politis and Romano (1994) D. N. Politis and J. P. Romano.  Large Sample Confidence Regions Based on Subsamples under Minimal Assumptions.  _The Annals of Statistics_ , 22(4):2031–2050, 1994.  doi: 10.1214/aos/1176325770.
  * Politis et al. (1999) D. N. Politis, J. P. Romano, and M. Wolf.  _Subsampling_.  Springer Series in Statistics, 1999.  ISBN 978-0-387-98854-2.  doi: 10.1007/978-1-4612-1554-7.
  * Rényi (1953) A. Rényi.  On The Theory Of Order Statistics.  _Acta Mathematica Academiae Scientiarum Hungarica_ , 4:191–231, 1953.  doi: 10.1007/BF02127580.
  * Resnick (1987) S. I. Resnick.  _Extreme Values, Regular Variation and Point Processes_.  Springer, 1987.  ISBN 978-0-387-75952-4.  doi: 10.1007/978-0-387-75953-1.
  * Sasaki and Wang (2022) Y. Sasaki and Y. Wang.  Fixed-k Inference for Conditional Extremal Quantiles.  _Journal of Business and Economic Statistics_ , 40(2):829–837, 2022.  ISSN 15372707.  doi: 10.1080/07350015.2020.1870985.
  * Serfling (1980) R. J. Serfling.  _Approximation Theorems of Mathematical Statistics_.  John Wiley & Sons, Ltd, 1980.  ISBN 9780471024033.  doi: 10.1002/9780470316481.
  * Zhang (2018) Y. Zhang.  Extremal Quantile Treatment Effects.  _The Annals of Statistics_ , 46(6):3707–3740, 2018.  doi: 10.1214/17-AOS1673.

## Proofs of Results in the Main Text

###  Appendix A Distributional Results

####  A.1 Proof of Theorem 3.1

We turn to the proof of theorem 3.1. We begin by stating some auxiliary
results. Let $H_{T}$ be the cdf of
$\vartheta_{i,T}=\theta_{i}+T^{-p}\varepsilon_{i,T}$. Observe that
$\\{\vartheta_{i,T}\\}$ form a triangular array with rows indexed by $T$, and
number of entries in each row given by $N$. In each row, the entries are iid
and distributed according to the cdf $H_{T}$. Define the auxiliary functions
$U_{T}$ and $U_{F}$ as

$U_{T}=\left(\dfrac{1}{1-H_{T}}\right)^{-1},\quad
U_{F}=\left(\dfrac{1}{1-F}\right)^{-1}.$ (A.1)

Using $U_{T}$ and $U_{F}$ greatly simplifies notation in subsequent proofs.
Observe the following useful connection between $U_{T}$ and $H_{T}$. Let
$\tau\in(0,\infty)$. Let $N$ be large enough so that $N\tau>1$. Then

$\displaystyle U_{T}(N\tau)$
$\displaystyle=H_{T}^{-1}\left(1-\dfrac{1}{N\tau}\right),\quad
U_{F}(N\tau)=F^{-1}\left(1-\dfrac{1}{N\tau}\right).$ (A.2)

We begin with a technical lemma that connects convergence of the normalized
sample maximum
$\vartheta_{N,N,T}=\max\left\\{\vartheta_{1,T},\dots,\vartheta_{N,T}\right\\}$
and convergence of the quantiles of $H_{T}$.

######  Lemma A.1.

Let assumption 1 hold. The following are equivalent:

  1. (1)

As $N,T\to\infty$, for some constants $a_{N},b_{N}$ the random variable
${(\vartheta_{N,N,T}-b_{N})}/{a_{N}}$ converges weakly to a random variable
$X$ with non-degenerate cdf $Q(x)$.

  2. (2)

As $N,T\to\infty$, for all $\tau\in(0,\infty)$ such that $\tau$ is a
continuity of point $Q^{-1}(\exp({-1/\tau}))$, it holds that
$(U_{T}(N\tau)-b_{N})/{a_{N}}\Rightarrow Q^{-1}(\exp({-1/\tau}))$.

The same is also true for $\theta_{N,N}$ with $U_{F}$ in place of $U_{T}$.

The proof is completely analogous to the proof of theorem 1.1.2 in de Haan and
Ferreira (2006), which establishes a similar result for the iid case. We
provide the full argument for completeness.

###### Proof.

First consider the implication $(1)\Rightarrow(2)$. Under assumption 1 the
individual $\vartheta_{i,T}$ are iid RVs with cdf $H_{T}$, hence
$P\left(({\vartheta_{N,N,T}-b_{N}})/{a_{N}}\leq
x\right)=H_{T}^{N}(a_{N}x+b_{N})$. By assumption, $H_{T}^{N}(a_{N}x+b_{N})$
converges to some nondegenerate cdf $Q(x)$ as $N,T\to\infty$ at all points of
continuity of $Q(x)$. Let $x$ be a continuity point of $Q(x)$ such that
$Q(x)\in(0,1)$. Then at such an $x$ take logs

$N\log H_{T}(a_{N}x+b_{N})\Rightarrow\log Q(x).$ (A.3)

For any such $x$ it must be that $H_{T}(a_{N}x+b_{N})\to 1$, otherwise the
left hand side will diverge to $-\infty$. Since $\log(1+x)\sim x$ for $x\sim
0$, this implies that

$\lim_{N,T\to\infty}\frac{-\log
H_{T}(a_{N}x+b_{N})}{1-H_{T}(a_{N}x+b_{N})}=1.$ (A.4)

Replacing the logarithm above, we obtain
$N\left(1-H_{T}(a_{N}x+b_{N})\right)\Rightarrow-\log Q(x)$. Taking reciprocals
of both sides yields

$\dfrac{1}{N\left(1-H_{T}(a_{N}x+b_{N})\right)}\Rightarrow\dfrac{1}{-\log
Q(x)}.$ (A.5)

Taking inverses in eq. (A.5) and using the definition (A.1) of $U_{T}$, we
obtain that weak convergence of the sample maximum is equivalent to the
following convergence of quantiles of $H_{T}$:

$\dfrac{U_{T}(Ny)-b_{N}}{a_{N}}\Rightarrow Q^{-1}(e^{-1/y})\text{ as
}N,T\to\infty\leavevmode\nobreak\ .$ (A.6)

Following the above argument in the reverse direction establishes
$(2)\Rightarrow(1)$. ∎

The following lemma provides a condition on $U_{T}$ under which
$\vartheta_{N,N,T}$ and $\theta_{N,N}$ have the same limit properties.

######  Lemma A.2.

If

  1. (1)

Assumption 1 holds

  2. (2)

$a_{N},b_{N}$ are such that as $N\to\infty$ the normalized noiseless maximum
$({\theta_{N,N}-b_{N}})/a_{N}$ converges weakly to a non-degenerate random
variable $X$.

  3. (3)

For each $\tau\in(0,\infty)$ it holds that
${(U_{T}(N\tau)-U_{F}(N\tau))}/{a_{N}}\to 0\text{ as }N,T\to\infty$

then as $N,T\to\infty$ ${(\vartheta_{N,N,T}-b_{N})}/{a_{N}}\Rightarrow X$.

###### Proof.

Let $Q$ be the cdf of $X$. Let $\tau\in(0,\infty)$ be a continuity point of
$Q^{-1}(\exp(-1/\tau))$. Then by lemma A.1 $a_{N}^{-1}({U_{F}(N\tau)-b_{N}})$
converges to $Q^{-1}(\exp(-1/\tau))$. By assumption (3) of the lemma for all
$\tau\in(0,\infty)$ continuity points of $Q^{-1}(\exp(-1/\tau))$ it holds that

$\dfrac{U_{T}(N\tau)-{b}_{N}}{a_{N}}-\dfrac{U_{F}(N\tau)-b_{N}}{a_{N}}=\dfrac{U_{T}(N\tau)-U_{F}(N\tau)}{a_{N}}\to
0\text{ as }N,T\to\infty.$ (A.7)

From this we conclude that $a_{N}^{-1}({U_{T}(N\tau)-b_{N}})\to
Q^{-1}(\exp(-1/\tau))$ for all $\tau\in(0,\infty)$ continuity points of
$Q^{-1}(\exp(-1/\tau))$. The result follows from lemma A.1. ∎

We will make use of the following quantile inequalities due to Makarov (1981).

######  Lemma A.3 (Makarov quantile inequalities; eqs. (1) and (2) in Makarov
(1981)).

Suppose that $X\sim F_{X}$ and $Y\sim F_{Y}$ are a pair of random variables
whose joint distribution is not restricted, and consider their sum $X+Y\sim
F_{X+Y}$. The following inequalities hold: for all $v\in[0,1]$

$\displaystyle F^{-1}_{X+Y}(v)$
$\displaystyle\leq\inf_{w\in[v,1]}\left(F_{X}^{-1}(w)+F^{-1}_{Y}(1+v-w)\right),$
(A.8) $\displaystyle F^{-1}_{X+Y}(v)$
$\displaystyle\geq\sup_{w\in[0,v]}\left(F_{X}^{-1}(w)+F_{Y}^{-1}(v-w)\right).$
(A.9)

The bounds are pointwise sharp in the following sense: for each $v$ there
exists a joint distribution of $X$ and $Y$ such that the $v$th quantile of
$X+Y$ attains the lower/upper bound at $v$.

###### Proof of theorem 3.1.

Let $H_{T}$ be the cdf of $\vartheta_{i,T}$. Fix $\tau\in(0,\infty)$. Let $N$
be large enough so that $N\tau>1$. In the statement of Makarov’s inequalities
(lemma A.3), take $X=\theta_{i}$ and $Y=T^{-p}\varepsilon_{i,T}$, and
$v=1-1/N\tau$ and subtract $F^{-1}(1-1/N\tau)$ on all sides to obtain

$\displaystyle\sup_{w\in\left[0,1-\frac{1}{N\tau}\right]}\left(F^{-1}(w)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(1-\frac{1}{N\tau}-w\right)-F^{-1}\left(1-\dfrac{1}{N\tau}\right)\right)$
(A.10) $\displaystyle\leq$ $\displaystyle
H^{-1}_{T}\left(1-\frac{1}{N\tau}\right)-F^{-1}\left(1-\frac{1}{N\tau}\right)$
(A.11) $\displaystyle\leq$
$\displaystyle\inf_{w\in\left[1-\frac{1}{N\tau},1\right]}\left(F^{-1}(w)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(1+1-\frac{1}{N\tau}-w\right)-F^{-1}\left(1-\dfrac{1}{N\tau}\right)\right)\leavevmode\nobreak\
.$ (A.12)

First, by definitions of $U_{T}$ and $U_{F}$ (eq. (A.1)) and eq. (A.2), the
middle of eq (A.11) is equal to $U_{T}(N\tau)-U_{F}(N\tau)$. Second, in the
supremum condition define $u=1-1/(N\tau)-w$ to write the condition as

$\sup_{u\in\left[0,1-\frac{1}{N\tau}\right]}\left(F^{-1}\left(1-\dfrac{1}{N\tau}-u\right)-F^{-1}\left(1-\dfrac{1}{N\tau}\right)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(u\right)\right)$
(A.13)

Note that eventually $[0,\epsilon]\subset[0,1-1/(N\tau)]$, and so the
expression in eq. (A.11) may further be lower-bounded as
$\sup_{u\in[0,\epsilon]}\\{\dots\\}\leq\sup_{u\in[0,1-1/(N\tau)]}\\{\dots\\}$.
Third, define $u=-[1-1/(N\tau)-w]$ in the infimum condition in eq. (A.11) to
represent it as (TE-Inf). Suppose that $a_{N}>0$ (if not, simply reverse all
inequalities below). Combining the above arguments and multiply all sides by
$a_{N}^{-1}$, we obtain

$\displaystyle\sup_{u\in\left[0,\epsilon\right]}\dfrac{1}{a_{N}}\left(F^{-1}\left(1-\dfrac{1}{N\tau}-u\right)-F^{-1}\left(1-\dfrac{1}{N\tau}\right)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(u\right)\right)$
(A.14) $\displaystyle\leq$
$\displaystyle\dfrac{U_{T}(N\tau)-U_{F}(N\tau)}{a_{N}}$ (A.15)
$\displaystyle\leq$
$\displaystyle\inf_{u\in\left[0,\frac{1}{N\tau}\right]}\dfrac{1}{a_{N}}\left(F^{-1}\left(1-\dfrac{1}{N\tau}+u\right)-F^{-1}\left(1-\dfrac{1}{N\tau}\right)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(1-u\right)\right)\leavevmode\nobreak\
.$ (A.16)

Then conditions (TE-Inf) and (TE-Sup) imply that
$(U_{T}(N\tau)-U_{F}(N\tau))/{a_{N}}\to 0$ for all $\tau\in(0,\infty)$. By
lemma A.2 it follows that $(\vartheta_{N,N,T}-b_{N})/{a_{N}}\Rightarrow X$.

Now we turn to the second assertion. First, the above shows that (TE-Sup) (TE-
Inf) together imply (TE-Sup’). Now suppose that at least one of (TE-Inf) and
(TE-Sup’) fails. We show that the limit properties of
$a_{N}^{-1}({\vartheta_{N,N,T}-b_{N}})$ differ from the limit properties of
$a_{N}^{-1}({\theta_{N,N}-b_{N}})$ for some sequence of joint distributions of
$(\theta_{i},\varepsilon_{i,T})$. Suppose it is the infimum condition (TE-Inf)
that fails to hold for some $\tau$; the argument for (TE-Sup’) is identical.
Then along some subsequence of $(N,T)$ it holds that
$\inf_{u\in\left[1-{1}/{N\tau},1\right]}a_{N}^{-1}\left(\cdots\right)=\delta_{N,T}$
such that $\delta_{N,T}$ are bounded away from zero. Suppose that it is
possible to extract a further subsequence such that along it $\delta_{N,T}$
converges to some $\delta\neq 0$. Pass to that subsubsequence. Theorem 2 in
Makarov (1981) establishes that for each $(N,T)$ there exists a joint
distribution of $\theta_{i}$ and $\varepsilon_{i,T}$ such that the resulting
$H_{T}$ attains the upper bound in inequality (A.11), and so

$\displaystyle H^{-1}_{T}\left(1-\dfrac{1}{N\tau}\right)$
$\displaystyle=a_{N}\delta_{N,T}+F^{-1}\left(1-\frac{1}{N\tau}\right).$ (A.17)

If $({U_{F}(N\tau)-b_{N}})/{a_{N}}\to Q(\exp(-1/\tau))$, then for such a
sequence of $H_{T}$ it holds that $(U_{T}(N\tau)-b_{N})/{a_{N}}\to
Q^{-1}(\exp(-1/\tau))+\delta$. Suppose that $(\vartheta_{N,N,T}-b_{N})/a_{N}$
converges to a random variable with distribution $\tilde{Q}$ along the same
subsequence. The above discussion shows that
$\tilde{Q}^{-1}(\exp(-1/\tau))=Q^{-1}(\exp(-1/\tau))+\delta\neq
Q^{-1}(\exp(-1/\tau))$. Thus, either the limit distribution of
$\vartheta_{N,N,T}$ is different from that of $\theta_{N,N}$, or
$\vartheta_{N,N,T}$ does not converge.

If we cannot extract a subsequence of $\delta_{N,T}$ converging to some finite
$\delta$, then $\delta_{N,T}$ is unbounded. In this case it is possible to
extract a further monotonically increasing subsequence. Proceeding as above,
we obtain that along that subsequence it holds that
${(U_{T}(N\tau)-b_{N})}/{a_{N}}\to Q^{-1}(\exp(-1/\tau))+\infty$, and so
$\left(\vartheta_{N,N,T}-b_{N}\right)/a_{N}$ does not converge. ∎

####  A.2 Proof of Proposition 3.2

Before proving proposition 3.2, we state two useful results. First, let
$RV_{\gamma}$ be the class of non-negative functions of regular variation with
parameter $\gamma$, that is, those measurable
$f:\operatorname{\mathbb{R}}_{+}\to\operatorname{\mathbb{R}}_{+}$ that satisfy
$\lim_{t\to\infty}{f(tx)}/{f(t)}=x^{\gamma}$ for any $x>0$. $RV_{0}$ is the
class of slowly varying functions.

######  Lemma A.4 (Karamata Characterization Theorem; Theorem 1.4.1 in Bingham
et al. (1987)).

Let $f\in RV_{\gamma}$. Then there exists a slowly varying function $L$ (that
is, $L\in RV_{0}$) such that for all $x$ it holds that $f(x)=x^{\gamma}L(x)$.

Second, assumption 2 is equivalent to the following statement: for some
sequences $\alpha_{N},\beta_{N}$ and all $x>0$

$\lim_{N\to\infty}\dfrac{U_{F}(Nx)-U_{F}(N)}{\alpha_{N}}=\dfrac{x^{\gamma}-1}{\gamma}\text{
(meaning $\log(x)$ for $\gamma=0$)},$ (A.18)

Since the left hand side is monotonic in $x$ and the right hand side is
continuous, convergence in (A.18) is locally uniform in $x$ (that is, for any
$0<a,b<\infty$ convergence in (A.18) is uniform on $[a,b]$). See theorem 1.1.6
and corollary 1.2.4 in de Haan and Ferreira (2006). We call the constants the
$\alpha_{N},\beta_{N}$ _canonical_ normalization constants.

###### Proof of proposition 3.2.

Let $(a_{N},b_{N})$ be such that $(\theta_{N,N}-b_{N})/a_{N}\Rightarrow X$ for
some non-degenerate random variable $X$. Fix $\tau\in(0,\infty)$ and define
$s_{\tau,N,T}$ and $S_{\tau,N,T}$ as

$\displaystyle S_{\tau,N,T}(u)$
$\displaystyle=\dfrac{1}{a_{N}}\left(F^{-1}\left(1-\dfrac{1}{N\tau}+u\right)-F^{-1}\left(1-\dfrac{1}{N\tau}\right)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(1-u\right)\right),$
(A.19) $\displaystyle s_{\tau,N,T}(u)$
$\displaystyle=\dfrac{1}{a_{N}}\left(F^{-1}\left(1-\dfrac{1}{N\tau}-u\right)-F^{-1}\left(1-\dfrac{1}{N\tau}\right)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(u\right)\right).$
(A.20)

Condition (TE-Inf) for $F$ and $G_{T}$ can be written as
$\inf_{u\in\left[0,{1}/{N\tau}\right]}S_{\tau,N,T}(u)\to 0$, while condition
(TE-Sup) can be written as
$\sup_{u\in\left[0,\epsilon\right]}s_{\tau,N,T}(u)\to 0$ for some
$\epsilon\in(0,1)$

By lemma A.3 and the discussion after eq. (A.11)

$\displaystyle\sup_{u\in\left[0,1-\frac{1}{N\tau}\right]}\left[F^{-1}\left(1-\dfrac{1}{N\tau}-u\right)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(u\right)\right]\leq\inf_{u\in\left[0,\frac{1}{N\tau}\right]}\left[F^{-1}\left(1-\dfrac{1}{N\tau}+u\right)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(1-u\right)\right].$

Suppose that $a_{N}>0$ (if not, simply reverse all inequalities below). The
above inequality implies that
$\sup_{u\in\left[0,1-{1}/{N\tau}\right]}s_{\tau,N,T}(u)\leq\inf_{u\in\left[0,{1}/{N\tau}\right]}S_{\tau,N,T}(u)$.
Further, fix an arbitrary $\epsilon\in(0,1)$. Then eventually
$[0,\epsilon]\subset[0,1-1/N\tau]$, and correspondingly
$\sup_{u\in\left[0,\epsilon\right]}s_{\tau,N,T}(u)\leq\sup_{u\in\left[0,1-{1}/{N\tau}\right]}s_{\tau,N,T}(u)$.
If $u_{s,\tau,N,T}\in[0,\epsilon]$ and $u_{S,\tau,N,T}\in[0,1/N\tau]$, then
the following chain of inequalities holds:

$s_{\tau,N,T}(u_{s,\tau,N,T})\leq\sup_{u\in\left[0,\epsilon\right]}s_{\tau,N,T}(u)\leq\inf_{u\in\left[0,\frac{1}{N\tau}\right]}S_{\tau,N,T}(u)\leq
S_{\tau,N,T}(u_{S,\tau,N,T})\leavevmode\nobreak\ .$ (A.21)

Let $F$ satisfy assumption 2 with EV index $\gamma>\gamma^{\prime}$. Define

$u_{S,\tau,N,T}=\frac{1}{N\tau}\frac{1}{\log(T)+1}\in\left[0,\dfrac{1}{N\tau}\right].$
(A.22)

Suppose that under the conditions of the proposition it holds that

$\displaystyle\dfrac{1}{a_{N}}\left(F^{-1}\left(1-\dfrac{1}{N\tau}+u_{S,\tau,N,T}\right)-F^{-1}\left(1-\dfrac{1}{N\tau}\right)\right)$
$\displaystyle\to 0,$ (A.23)
$\displaystyle\dfrac{1}{a_{N}}\dfrac{1}{T^{p}}G_{T}^{-1}\left(1-u_{S,\tau,N,T}\right)$
$\displaystyle\to 0.$ (A.24)

Under (A.23) and (A.24) it holds that $S_{\tau,N,T}(u_{S,\tau,N,T})\to 0$ as
$N,T\to\infty$. By inequality (A.21) we conclude that
$\limsup_{N,T\to\infty}\inf_{u\in\left[0,{1}/{N\tau}\right]}S_{\tau,N,T}(u)\leq
0$. An identical argument shows that $s_{\tau,N,T}(u_{s,\tau,N,T})\to 0$ where
$u_{s,\tau,N,T}=({1}/{N\tau})(1/{\log(T)})$ eventually lies in $[0,\epsilon]$
for any $\epsilon\in(0,1)$; hence
$\liminf_{N,T\to\infty}\sup_{u\in\left[0,\epsilon\right]}s_{\tau,N,T}(u)\geq
0$. Further,
$\limsup_{N,T\to\infty}\sup_{u\in\left[0,\epsilon\right]}s_{\tau,N,T}(u)\leq\liminf_{N,T\to\infty}\inf_{u\in\left[0,{1}/{N\tau}\right]}S_{\tau,N,T}(u)$
by inequality (A.21). Combining the results, we obtain that both
$\inf_{u\in\left[0,{1}/{N\tau}\right]}S_{\tau,N,T}(u)$ and
$\sup_{u\in\left[0,\epsilon\right]}s_{\tau,N,T}(u)$ tend to 0, proving the
result.

It remains to establish (A.23) and (A.24). We split the proof of (A.23) by
sign of $\gamma$.

Suppose $\gamma>0$. We begin by making two observations. First, by the
convergence to types theorem (Resnick, 1987, proposition 0.2)
$a_{N}\sim\alpha_{N}$ for the canonical scaling $\alpha_{N}$ of eq. (A.18).
Further, by lemma 1.2.9 of de Haan and Ferreira (2006) $\alpha_{N}\sim
U_{F}(N)$. Second, by corollary 1.2.10 in de Haan and Ferreira (2006)
$(U_{F}(x))^{-1}\in RV_{-\gamma}$. Then by lemma A.4 we can write
$1/U_{F}(x)=x^{-\gamma}L(x)$ where $L$ is a slowly varying function (that
depends on $F$). Combining the above observations, definition of
$u_{S,\tau,N,T}$ and eq. (A.2), we see that

$\displaystyle\dfrac{1}{a_{N}}\left(F^{-1}\left(1-\frac{1}{N\tau}\frac{\log(T)}{\log(T)+1}\right)-F^{-1}\left(1-\dfrac{1}{N\tau}\right)\right)$
(A.25)
$\displaystyle\sim\dfrac{1}{U_{F}(N)}\left(U_{F}\left(N\tau\dfrac{\log(T)+1}{\log(T)}\right)-U_{F}(N\tau)\right)$
(A.26)
$\displaystyle=N^{-\gamma}L(N)\left(\left(N\tau\dfrac{\log(T)+1}{\log(T)}\right)^{\gamma}\dfrac{1}{L\left(N\tau\frac{\log(T)+1}{\log(T)}\right)}-(N\tau)^{\gamma}\dfrac{1}{L(N\tau)}\right)$
(A.27)
$\displaystyle\propto\left(\dfrac{\log(T)+1}{\log(T)}\right)^{\gamma}\dfrac{L(N)}{L\left(N\tau\frac{\log(T)+1}{\log(T)}\right)}-\dfrac{L(N)}{L(N\tau)}\to
0.$ (A.28)

Convergence follows since $L$ is slowly varying on infinity:
${L(N)}/{L(N\tau)}\to 1$. By local uniform convergence (proposition 0.5 in
Resnick (1987)) ${L(N)}/{L\left(N\tau{(\log(T)+1)}/{\log(T)}\right)}\to 1$.

Suppose $\gamma<0$. As above, $a_{N}\sim\alpha_{N}$. By lemma 1.2.9 in de Haan
and Ferreira (2006) in turn $\alpha_{N}\sim(U_{F}(\infty)-U_{F}(N))$ (note
that $U_{F}(\infty)<1$ when $\gamma<0$). By corollary 1.2.10 in de Haan and
Ferreira (2006), $(U_{F}(\infty)-U(x))^{-1}\in RV_{-\gamma}$ and we can write
$1/(U_{F}(\infty)-U(x))=x^{-\gamma}L(x)$ for some slowly varying function $L$
by lemma A.4. Hence proceeding as for $\gamma>0$

$\displaystyle\dfrac{1}{a_{N}}\left(F^{-1}\left(1-\frac{1}{N\tau}\frac{\log(T)}{\log(T)+1}\right)-F^{-1}\left(1-\dfrac{1}{N\tau}\right)\right)$
(A.29)
$\displaystyle\sim\dfrac{1}{U_{F}(\infty)-U_{F}(N)}\left(\left(U_{F}\left(N\tau\dfrac{\log(T)+1}{\log(T)}\right)-U_{F}(\infty)\right)-(U_{F}(N\tau)-U(\infty))\right)$
(A.30)
$\displaystyle=N^{-\gamma}L(N)\left(\left(N\tau\dfrac{\log(T)+1}{\log(T)}\right)^{\gamma}\dfrac{1}{L\left(N\tau\frac{\log(T)+1}{\log(T)}\right)}-(N\tau)^{\gamma}\dfrac{1}{L(N\tau)}\right)$
(A.31)
$\displaystyle\propto\left(\dfrac{\log(T)+1}{\log(T)}\right)^{\gamma}\dfrac{L(N)}{L\left(N\tau\frac{\log(T)+1}{\log(T)}\right)}-\dfrac{L(N)}{L(N\tau)}\to
0.$ (A.32)

Suppose $\gamma=0$. As above, $a_{N}\sim\alpha_{N}$. By eq. (A.18)
$\lim_{N\to\infty}{(U_{F}(Nx)-U_{F}(N))}/{\alpha_{N}}=\log(x)$ locally
uniformly in $x$. Then

$\displaystyle\dfrac{1}{a_{N}}\left(F^{-1}\left(1-\frac{1}{N\tau}\frac{\log(T)}{\log(T)+1}\right)-F^{-1}\left(1-\dfrac{1}{N\tau}\right)\right)$
(A.33)
$\displaystyle\sim\dfrac{1}{\alpha_{N}}\left(U_{F}\left(N\tau\frac{\log(T)+1}{\log(T)}\right)-U_{F}(N\tau)\right)\to\log\left(\lim_{T\to\infty}\dfrac{\log(T)+1}{\log(T)}\right)=0.$
(A.34)

Now we focus on the $G^{-1}_{T}$ term in eq. (A.24). First suppose that
$\sup_{T}\operatorname{\mathbb{E}}\left\lvert\varepsilon_{i,T}\right\rvert^{\beta}<\infty$.
Then by Markov’s inequality we obtain for any $\tau\in(0,1)$ that
$G_{T}^{-1}(\tau)\leq\left({\sup_{T}\operatorname{\mathbb{E}}\lvert\varepsilon_{i,T}\rvert^{\beta}}/({1-\tau})\right)^{1/\beta}$.
Hence

$\frac{1}{T^{p}}G^{-1}_{T}\left(1-\frac{1}{N\tau}\frac{1}{\log(T)+1}\right)=O\left(\frac{N^{1/\beta}(\log(T))^{1/\beta}}{T^{p}}\right)$
(A.35)

where the $O$ term is uniform in $T$. First suppose that $\gamma\neq 0$. As
above, $a_{N}^{-1}\sim x^{-\gamma}L(x)$ where $L$ is a slowly varying function
(that depends on $F$). For eq. (A.24) to hold it is sufficient that
${N^{1/\beta-\gamma}L(x)(\log(T))^{1/\beta}}/{T^{p}}\to 0$ (by eq. (A.35)).
Write $\gamma=\gamma^{\prime}+\delta$, $\delta>0$. Then

$\frac{N^{1/\beta-\gamma}L(x)(\log(T))^{1/\beta}}{T^{p}}=\left[\frac{N^{1/\beta-\gamma^{\prime}}(\log(T))^{1/\beta}}{T^{p}}\right]\left[\frac{L(x)}{N^{\delta}}\right]\to
0$ (A.36)

since the condition holds for $\gamma^{\prime}$ and ${L(x)}/{x^{\delta}}\to 0$
for any $\delta>0$ ($L$ is slowly varying). Second, let $\gamma=0$. Then
(A.24) holds if ${N^{1/\beta}(\log(T))^{1/\beta}}/(\alpha_{N}T^{p})\to 0$. Fix
an arbitrary $x>0$ and write this as

$\displaystyle\dfrac{N^{1/\beta}(\log
T)^{1/\beta}}{\alpha_{N}T^{p}}=\dfrac{N^{-1/\beta-\gamma^{\prime}}(\log(T))^{1/\beta}}{T^{p}}\dfrac{U_{F}(Nx)-U_{F}(N)}{\alpha_{N}}\dfrac{1}{N^{-\gamma^{\prime}}(U_{F}(Nx)-U_{F}(N))}$
(A.37)

By assumption, the first fraction tends to zero. The second fraction tends to
$\log(x)$ by eq. (A.18). Last, there are two possibilities for
$U_{F}(Nx)-U_{F}(N)$. The first one is that it is bounded away from zero. The
second one is that it converges to zero (possibly along a subsequence); in
this case convergence is slower than $N^{-\kappa}$ for all $\kappa>0$ by
problem 1.1.1(b) in Resnick (1987) (recall that $U_{F}(N)=F^{-1}(1-1/N)$ by
eq. (A.2)). In both cases the last fraction converges to zero, as
$\gamma^{\prime}<0$.

The proof is identical if $G_{T}\sim N(\mu_{T},\sigma_{T}^{2})$. Assumption 3
implies that $\mu_{T}$ and $\sigma_{T}^{2}$ are bounded. In this case

$\dfrac{1}{T^{p}}G^{-1}_{T}\left(1-\dfrac{1}{N\tau}\left(1-\frac{\log(T)}{\log(T)+1}\right)\right)=O\left(\dfrac{\sqrt{\log(N)}}{T^{p}}\right),$
(A.38)

where the $O$ term is uniform in $T$. The rest of the argument proceeds as
above. ∎

####  A.3 Proof of Theorem 3.3

###### Proof of theorem 3.3.

By theorem 2.2.1 in de Haan and Ferreira (2006)

$\sqrt{k}\dfrac{\theta_{N-k,N}-U_{F}\left(\frac{N}{k}\right)}{\frac{N}{k}U^{\prime}_{F}\left(\frac{N}{k}\right)}\Rightarrow
N(0,1).$ (A.39)

where
$({N}/{k})U^{\prime}_{F}\left({N}/{k}\right)=(N/k)\times\left(\left({1}/({1-F})\right)^{-1}\right)^{\prime}\left(N/k\right)\equiv
c_{N}$ (by eqs. (A.1) and (A.2)). We now transfer this convergence property to
$\vartheta_{N-k,N}$.

It is convenient for the purposes of the proof to replace the uniform random
variables $U_{i}$ with $1/U_{i}$. Let $Y_{1},\dots,Y_{N}$ be iid random
variables with cdf $F_{Y}(y)=1-1/y,y>1$, $F_{Y}(y)=0$ for $y\leq 1$. Observe
that $Y_{i}\overset{d}{=}1/U_{i}$ where $U_{i}$ is Uniform[0, 1]. Let
$Y_{1,N}\leq\dots\leq Y_{N,N}$ be the order statistics, then
$U_{k,N}\overset{d}{=}1/Y_{N-k,N}$. As pointed out by de Haan and Ferreira
(2006) (p. 50)

$\displaystyle\theta_{N-k,N}$
$\displaystyle\overset{d}{=}U_{F}(Y_{N-k,N}),\quad\vartheta_{N-k,N,T}\overset{d}{=}U_{T}(Y_{N-k,N}).$
(A.40)

Let $c_{N}=({N}/{k})U^{\prime}_{F}\left({N}/{k}\right)$, as in the statement
of the theorem. Then

$\displaystyle\sqrt{k}\dfrac{\vartheta_{N-k,N,T}-U_{F}\left(\frac{N}{k}\right)}{c_{N}}$
$\displaystyle\overset{d}{=}\sqrt{k}\dfrac{U_{T}\left(Y_{N-k,N}\right)-U_{F}\left(\frac{N}{k}\right)}{c_{N}}$
(A.41)
$\displaystyle=\sqrt{k}\dfrac{U_{F}\left(Y_{N-k,N}\right)-U_{F}\left(\frac{N}{k}\right)}{c_{N}}+\dfrac{\sqrt{k}}{c_{N}}\left(U_{T}\left(Y_{N-k,N}\right)-U_{F}\left(Y_{N-k,N}\right)\right)$
(A.42)

By eqs. (A.39) and (A.40) it follows that as $N\to\infty$ the first term
converges weakly to a $N(0,1)$ variable. The conclusion of the theorem follows
if

$\dfrac{\sqrt{k}}{c_{N}}\left(U_{T}\left(Y_{N-k,N}\right)-U_{F}\left(Y_{N-k,N}\right)\right)\xrightarrow{p}0\leavevmode\nobreak\
.$ (A.43)

We establish eq. (A.43) by an argument similar to the one used in the proof of
theorem 3.1. We use lemma A.3 to bound
$U_{T}(Y_{N-k,N})=F^{-1}\left(1-1/Y_{N-k,N}\right)$. In lemma A.3 take
$X=\theta_{i}$, $Y=T^{-p}\varepsilon_{i,T}$, $v=1-1/Y_{N-k,N}$, subtract
$U_{F}(Y_{N-k,n})=F^{-1}\left(1-1/Y_{N-k,N}\right)$ on all sides and multiply
by $\sqrt{k}/c_{N}$ to obtain

$\displaystyle\sup_{w\in\left[0,1-\frac{1}{Y_{N-k,N}}\right]}\dfrac{\sqrt{k}}{c_{N}}\left(F^{-1}(w)+\dfrac{1}{T^{p}}G^{-1}_{T}\left(1-\frac{1}{Y_{N-k,N}}-w\right)-F^{-1}\left(1-\dfrac{1}{Y_{N-k,N}}\right)\right)$
(A.44)
$\displaystyle\leq\dfrac{\sqrt{k}}{c_{N}}\left(U_{T}(Y_{N-k,N})-U_{F}(Y_{N-k,N})\right)$
(A.45)
$\displaystyle\leq\inf_{w\in\left[1-\frac{1}{Y_{N-k,N}},1\right]}\dfrac{\sqrt{k}}{c_{N}}\left(F^{-1}(w)+\dfrac{1}{T^{p}}G^{-1}_{T}\left(1+1-\frac{1}{Y_{N-k,N}}-w\right)-F^{-1}\left(1-\dfrac{1}{Y_{N-k,N}}\right)\right),$
(A.46)

if $c_{N}$ is non-negative; the opposite inequalities hold if $c_{N}$ is
negative. Since $1/Y_{N-k,N}\overset{d}{=}U_{k,N}$, we obtain

$\displaystyle\inf_{w\in\left[1-\frac{1}{Y_{N-k,N}},1\right]}\dfrac{\sqrt{k}}{c_{N}}\left(F^{-1}(w)+\dfrac{1}{T^{p}}G^{-1}_{T}\left(1+1-\frac{1}{Y_{N-k,N}}-w\right)-F^{-1}\left(1-\dfrac{1}{Y_{N-k,N}}\right)\right)$
(A.47)
$\displaystyle\overset{d}{=}\inf_{w\in\left[1-U_{k,N},1\right]}\dfrac{\sqrt{k}}{c_{N}}\left(F^{-1}(w)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(1+1-U_{k,N}-w\right)-F^{-1}\left(1-U_{k,N}\right)\right)\xrightarrow{p}0$
(A.48)

Define $u=-[1-1/Y_{N-k,N}-w]$ to write the above as condition (3). For the
supremum, instead define $u=1-U_{k,N}-u$ and proceed as in the proof of
theorem 3.1, noting that $U_{k,N}\xrightarrow{p}0$ and hence with probability
approaching 1 $[0,\epsilon]\subset[0,1-U_{k,N}]$. Eq. (A.43) follows as in the
proof of theorem 3.1.

Sharpness of conditions (2) and (3) is established as in the proof of theorem
3.1. Suppose that condition (3) fails (the case for condition (2) is
analogous). There is some subsequence of $(N,T)$ and some $\delta_{N,T}$ such
that
$\inf_{u\in\left[1-U_{k,N},1\right]}\sqrt{k}c_{N}^{-1}(\cdot)=\delta_{N,T}$
and $\delta_{N,T}$ is bounded away from zero. Suppose that it is possible to
extract a further subsequence such that $\delta_{N,T}$ converges to some
$\delta\neq 0$. By theorem 2 in Makarov (1981), there exists a joint
distribution of $\theta_{i}$ and $\varepsilon_{i,T}$ such that the infimum is
attained. Then along this subsequence for this joint distribution
${\sqrt{k}}c_{N}^{-1}\left(U_{T}\left(Y_{N-k,N}\right)-U_{F}\left(Y_{N-k,N}\right)\right)\xrightarrow{p}\delta$.
Then from equations (A.39), (A.40), and (A.41) it follows that
$\sqrt{k}c_{N}^{-1}[\vartheta_{N-k,N,T}-U_{F}(N/k)]\Rightarrow N(\delta,1)$.
This convergence result may or may not hold for the overall original sequence.
If no convergent subsequence of $\delta_{N,T}$ can be extracted,
$\delta_{N,T}$ is unbounded. Extract a further monotonically increasing
subsequence. There exists a sequence of joint distributions of $\theta_{i}$
and $\varepsilon_{i,T}$ such that along that subsequence $\vartheta_{N-k,N,T}$
diverges. ∎

####  A.4 Proof of Proposition 3.4

###### Proof of proposition 3.4.

The proof proceeds similarly to that of proposition 3.2. As in the proof of
proposition 3.2, it is sufficient to prove that for some
$\tilde{u}_{S,N,T}\in\left[0,U_{k,N}\right]$ and $\tilde{u}_{s,N,T}$ that with
probability approaching 1 lies in $[0,\epsilon]$ for some $\epsilon\in(0,1)$.

$\displaystyle\dfrac{\sqrt{k}}{c_{N}}\left(F^{-1}(1-U_{k,N}+\tilde{u}_{S,N,T})-F^{-1}\left(1-U_{k,N}\right)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(1-\tilde{u}_{S,N,T}\right)\right)\xrightarrow{p}0,$
(A.49)
$\displaystyle\dfrac{\sqrt{k}}{c_{N}}\left(F^{-1}(1-U_{k,N}-\tilde{u}_{s,N,T})-F^{-1}\left(1-U_{k,N}\right)+\dfrac{1}{T^{p}}G_{T}^{-1}\left(\tilde{u}_{s,N,T}\right)\right)\xrightarrow{p}0.$
(A.50)

We only show that eq. (A.49) holds, eq. (A.50) follows analogously.

Let $\rho=\delta/2+\nu$, and set

$\tilde{u}_{S,N,T}=U_{k,N}\frac{1}{N^{\rho}+1}\in[0,U_{k,N}].$ (A.51)

As in the proof of proposition 3.2, we first show that the scaled $F^{-1}$
terms in eq. (A.49) decay, and then that the scaled $G^{-1}_{T}$ term decays.
First we establish that

$\dfrac{\sqrt{k}}{c_{N}}\left(F^{-1}\left(1-U_{k,N}\frac{N^{\rho}}{N^{\rho}+1}\right)-F^{-1}\left(1-U_{k,N}\right)\right)\xrightarrow{p}0.$
(A.52)

Let $Y_{1},\dots,Y_{N}$ be iid random variables with cdf $F_{Y}(y)=1-1/y,y>1$,
$F_{Y}(y)=0$ for $y\leq 1$. We will use that $Y_{i}\overset{d}{=}1/U_{i}$, and
correspondingly $Y_{N-k,N}\overset{d}{=}1/U_{k,N}$, as in the proof of theorem
3.3. Observe that $c_{N}$ can be written as $c_{N}=(N/k)U^{\prime}_{F}(N/k)$.
Then using eq. (A.2) we obtain that

$\displaystyle\dfrac{\sqrt{k}}{c_{N}}\left(F^{-1}\left(1-U_{k,N}\frac{N^{\rho}}{N^{\rho}+1}\right)-F^{-1}\left(1-U_{k,N}\right)\right)$
$\displaystyle\overset{d}{=}\dfrac{\sqrt{k}}{\frac{N}{k}U^{\prime}_{F}\left(\frac{N}{k}\right)}\left(U_{F}\left(Y_{N-k,N}\frac{N^{\rho}+1}{N^{\rho}}\right)-U_{F}(Y_{N-k,N})\right)$
$\displaystyle=\dfrac{\sqrt{k}}{\frac{N}{k}U^{\prime}_{F}\left(\frac{N}{k}\right)}\left(U_{F}\left(\frac{N}{k}\left(\frac{k}{N}Y_{N-k,N}\right)\frac{N^{\rho}+1}{N^{\rho}}\right)-U_{F}\left(\frac{N}{k}\left(\frac{k}{N}Y_{N-k,N}\right)\right)\right)$
$\displaystyle=\sqrt{k}\left(\dfrac{N^{\rho}+1}{N^{\rho}}-1\right)\left(\dfrac{k}{N}Y_{N-k,N}\right)\dfrac{\frac{N}{k}U_{F}^{\prime}\left(\frac{N}{k}x_{N}\right)}{\frac{N}{k}U^{\prime}_{F}\left(\frac{N}{k}\right)},\quad
x_{N}\in\left[\frac{k}{N}Y_{N-k,N},\left(\frac{k}{N}Y_{N-k,N}\right)\frac{N^{\rho}+1}{N^{\rho}}\right]$

where the last line follows by the mean value theorem. We now deal with the
last two terms in the above expression. By corollary 2.2.2 in de Haan and
Ferreira (2006), $({k}/{N})Y_{N-k,N}\xrightarrow{p}1$. By corollary 1.1.10 in
de Haan and Ferreira (2006) under assumptions 1 and 4 it holds that
$\lim_{t\to\infty}{U^{\prime}_{F}(tx)}/{U^{\prime}_{F}(t)}=x^{\gamma-1}$
locally uniformly in $x$. Since $x_{N}\to 1$ as $N\to\infty$ and
$k\to\infty,k=o(N)$, we conclude that
${U_{F}^{\prime}\left(\frac{N}{k}x_{N}\right)}/{U^{\prime}_{F}\left(\frac{N}{k}\right)}\to
1.$ Combining these observations, we obtain that

$\displaystyle\dfrac{\sqrt{k}}{\frac{N}{k}U^{\prime}_{F}\left(\frac{N}{k}\right)}\left(U_{F}\left(\frac{N}{k}\left(\frac{k}{N}Y_{N-k,N}\right)\frac{N^{\rho}+1}{N^{\rho}}\right)-U_{F}\left(\frac{N}{k}\left(\frac{k}{N}Y_{N-k,N}\right)\right)\right)$
(A.53)
$\displaystyle=O_{p}\left(\sqrt{k}\left[\dfrac{N^{\rho}+1}{N^{\rho}}-1\right]\right)=O_{p}(N^{\delta/2}N^{-\left(\delta/2+\nu\right)})=O_{p}(N^{-\nu})=o_{p}(1).$
(A.54)

where we apply the assumption that $k=N^{\delta}$ in the third equality.

Now we show that

$\dfrac{\sqrt{k}}{c_{N}}\dfrac{1}{T^{p}}G_{T}^{-1}\left(1-\tilde{u}_{S,N,T}\right)\xrightarrow{p}0.$
(A.55)

Suppose that
$\sup_{T}\operatorname{\mathbb{E}}\lvert\varepsilon_{i,T}\rvert^{\beta}<\infty$
(proof is analogous if $G_{T}\sim N(\mu_{T},\sigma^{2}_{T})$). As in eq.
(A.35)

$\dfrac{1}{T^{p}}G_{T}^{-1}\left(1-U_{k,N}\frac{1}{N^{\rho}+1}\right)\sim
O\left(\dfrac{N^{\rho/\beta}}{U_{k,N}^{1/\beta}T^{p}}\right)$ (A.56)

By corollary 1.1.10 in de Haan and Ferreira (2006) $U^{\prime}_{F}\in
RV_{\gamma-1}$, hence for some slowly varying function $L$ we can write
$U^{\prime}_{F}(x)=x^{\gamma-1}L(x)$ (be lemma A.4). Then
$c_{N}=({N}/{k})U_{F}^{\prime}\left({N}/{k}\right)=\left({N}/{k}\right)^{\gamma}L\left({N}/{k}\right)$.
Hence

$\displaystyle\dfrac{\sqrt{k}}{c_{N}}\dfrac{1}{T^{p}}G_{T}^{1}\left(1-\tilde{u}_{S,N,T}\right)=O\left(\sqrt{k}\frac{1}{\left(\frac{N}{k}\right)^{\gamma}L\left(\frac{N}{k}\right)}\dfrac{N^{\rho/\beta}}{U_{k,N}^{1/\beta}T^{p}}\right)$
(A.57)
$\displaystyle=O_{p}\left(k^{1/2+\gamma-1/\beta}N^{-\gamma+\rho/\beta+1/\beta}\dfrac{1}{L\left(\frac{N}{k}\right)}\dfrac{1}{T^{p}}\right)=O_{p}\left(\dfrac{N^{\delta/2(1+1/\beta)+(1-\delta)(-\gamma^{\prime}+1/\beta)+\nu/\beta}}{T^{p}}\dfrac{N^{-\varkappa(1-\delta)}}{L\left(N^{1-\delta}\right)}\right)$
(A.58)

where in the second equality we again use corollary 2.2.2 in de Haan and
Ferreira (2006) to conclude that $({N}/{k})\times
1/U_{k,N}\overset{d}{=}\left({k}/{N}\right)Y_{N-k,N}\xrightarrow{p}1$; in the
third line we write $\gamma=\gamma^{\prime}+\varkappa$ where $\varkappa>0$ by
assumption. The above expression is now $o_{p}(1)$ by assumptions of the
proposition and since $L$ is slowly varying.

Combining together equations (A.52) and (A.55) shows that (A.49) holds. To
prove that (A.50) holds, proceed as above
$\tilde{u}_{s,N,T}=U_{k,N}/N^{\rho}$; observe that $\tilde{u}_{s,N,T}$ lies in
$[0,\epsilon]$ with probability approaching 1 for any $\epsilon\in(0,1)$. ∎

###  Appendix B Inference

####  B.1 Proof of Lemmas 4.1\- 4.2, Theorems 4.3-4.4, Remark 9

###### Proof of lemma 4.1.

We split the proof by sign of $\gamma$. First, let $\gamma>0$. Let $Fr$ be an
RV with $P(Fr\leq x)=\exp(-x^{-1/\gamma})$ for $x\geq 0$ and $0$ for $x<0$;
note that $(E_{1}^{*})^{-\gamma}\overset{d}{=}Fr$. Then

$\displaystyle\dfrac{1}{F^{-1}\left(1-\frac{1}{N}\right)}\left[\vartheta_{N,N,T}-F^{-1}\left(1-\dfrac{l}{N}\right)\right]=\dfrac{1}{U_{F}(N)}\left[\vartheta_{N,N,T}-U_{F}\left(\dfrac{N}{l}\right)\right]$
(B.1) $\displaystyle=\dfrac{1}{U_{F}(N)}\vartheta_{N,N,T}-\dfrac{U_{F}(N\times
l^{-1})}{U_{F}(N)}\Rightarrow Fr-(l^{-1})^{\gamma}=Fr-\dfrac{1}{l^{\gamma}},$
(B.2)

since by corollary 1.2.10 in de Haan and Ferreira (2006) $U_{F}(x)\in
RV_{\gamma}$ and $\vartheta_{N,N,T}/U_{F}(N)\Rightarrow Fr$ by corollary 1.2.4
in de Haan and Ferreira (2006).

For $\gamma=0$ and let $Gu$ be an RV with $P(Gu\leq x)=\exp(-e^{-x})$ for
$x\in\operatorname{\mathbb{R}}$; note that $-\log(E_{1}^{*})\overset{d}{=}Gu$.
By corollary 1.2.4 in de Haan and Ferreira (2006) there exists a
$\hat{f}(\cdot)$ such that $\alpha_{N}=\hat{f}(F^{-1}(1-1/N))$ for
$\alpha_{N}$ of eq. (A.18). For this $\hat{f}$ we have

$\displaystyle\dfrac{1}{\hat{f}\left(F^{-1}\left(1-\frac{1}{N}\right)\right)}\left(\vartheta_{N,N,T}-F^{-1}\left(1-\frac{l}{N}\right)\right)$
$\displaystyle=\dfrac{1}{\alpha_{N}}\left(\vartheta_{N,N,T}-U_{F}(N)\right)+\dfrac{1}{\alpha_{N}}\left(U_{F}\left(\dfrac{N}{l}\right)-U_{F}(N)\right)\Rightarrow
Gu-\log(l),$

where the first term converges by corollary 1.2.4 in de Haan and Ferreira
(2006) and the second term by eq. (A.18).

For $\gamma<0$ necessarily $F^{-1}(1)=U_{F}(\infty)<\infty$. Let $W$ be an RV
with $P(W\leq x)=\exp(-(-x)^{-1/\gamma})$ for $x<0$ and $P(W\leq x)=1$ for
$x\geq 0$, note that $-(E_{1}^{*})^{-\gamma}\overset{d}{=}W$. Then

$\displaystyle\dfrac{1}{F^{-1}(1)-F^{-1}\left(1-\frac{1}{N}\right)}\left(\vartheta_{N,N,T}-F^{-1}\left(1-\frac{l}{N}\right)\right)$
(B.3)
$\displaystyle=\dfrac{1}{U_{F}(\infty)-U_{F}(N)}\left(\vartheta_{N,N,T}-U_{F}(\infty)\right)+\dfrac{U_{F}(\infty)-U_{F}(N/l)}{U_{F}(\infty)-U_{F}(N)}\Rightarrow
W+\frac{1}{l^{\gamma}},$ (B.4)

where the first term converges by corollary 1.2.4 in de Haan and Ferreira
(2006) and $U_{F}(\infty)-U_{F}(x)\in RV_{\gamma}$ by corollary 1.2.10 in de
Haan and Ferreira (2006). ∎

###### Proof of lemma 4.2.

Let $\alpha_{N}$ be as in eq. (A.18) and let $\beta_{N}=U_{F}(N)$. First we
show that for the constants $\alpha_{N},\beta_{N}$ it holds that

$\begin{pmatrix}\dfrac{\vartheta_{N,N,T}-\beta_{N}}{\alpha_{N}},\dfrac{\vartheta_{N-1,N,T}-\beta_{N}}{\alpha_{N}},\dots,\dfrac{\vartheta_{N-q,N,T}-\beta_{N}}{\alpha_{N}}\end{pmatrix}\\\
\Rightarrow\gamma^{-1}\begin{pmatrix}{(E_{1}^{*})^{-\gamma}-1},{(E_{1}^{*}+E_{2}^{*})^{-\gamma}-1},\dots,{(E_{1}^{*}+E_{2}^{*}+\dots+E^{*}_{q+1})^{-\gamma}-1}\end{pmatrix}.$
(B.5)

Let $E_{1},\dots,E_{q+1}$ be iid standard exponential RVs, and
$E_{1}^{*},\dots,E_{q+1}^{*}$ another iid set of standard exponential RVs.
Observe that $P\left(U_{T}\left(1/(1-\exp({-E_{i}}))\right)\leq
x\right)=H_{T}(x).$ Then

$\displaystyle(\vartheta_{N,N,T},\vartheta_{N-1,N,T},\dots,\vartheta_{N-q,N,T})$
$\displaystyle\overset{d}{=}$
$\displaystyle\left(U_{T}\left(\frac{1}{1-\exp({-E_{1,n}})}\right),U_{T}\left(\frac{1}{1-\exp{(-E_{2,n})}}\right),\dots,U_{T}\left(\frac{1}{1-\exp({-E_{q+1,n}})}\right)\right)$
$\displaystyle\overset{d}{=}$
$\displaystyle\left(U_{T}\left(\dfrac{1}{1-\exp\left({-\frac{E_{1}^{*}}{N}}\right)}\right),U_{T}\left(\dfrac{1}{1-\exp\left({-\frac{E_{1}^{*}}{N}-\frac{E_{2}^{*}}{N-1}}\right)}\right),\dots,\right.$
$\displaystyle\quad,\dots,\left.U_{T}\left(\dfrac{1}{1-\exp\left({-\frac{E_{1}^{*}}{N}-\frac{E_{2}^{*}}{N-1}-\dots-\frac{E_{q+1}^{*}}{N-q}}\right)}\right)\right),$

where the second equality follows by the Rényi (1953) representation of order
statistics from an exponential sample (see expression (1.9) in Rényi (1953)).
We conclude that

$\displaystyle\left(\dfrac{\vartheta_{N,N,T}-\beta_{N}}{\alpha_{N}},\dots,\dfrac{\vartheta_{N-q,N}-\beta_{N}}{\alpha_{N}}\right)$
(B.6) $\displaystyle\overset{d}{=}$