# Inference in Auctions with Many Bidders Using Transaction Prices††thanks: We
thank Joel Horowitz, Ulrich Müller, Rob Porter, and Daiqiang Zhang for helpful
comments, and Vinci Chow for kindly sharing the Hong Kong car license auction
data set. We also thank the participants of numerous seminars and conferences
for their helpful comments.

Federico A. Bugni
Department of Economics
Northwestern University
<EMAIL_ADDRESS>Yulong Wang
Department of Economics
Syracuse University
<EMAIL_ADDRESS>
###### Abstract

JEL code: C12, C57

##  1 Introduction

This paper considers inference in first-price and second-price sealed-bid
auctions with a large number of symmetric bidders having independent private
values (IPV). The abundance of bidders in each auction prompts us to consider
a novel asymptotic framework in which the number of bidders diverges, while
allowing the number of auctions to be small and remain fixed. This framework
differs substantially from the more conventional approach in which the
researcher observes multiple bids from a large number of independent and
identically distributed (i.i.d.) auctions. See Athey and Haile (2002); Haile
and Tamer (2003); Athey and Haile (2007); Guerre et al. (2000), among others.
Our analysis can deliver an accurate approximation in empirical settings in
which the number of bidders is large relative to the number of auctions. This
is especially suitable for applications with substantial heterogeneity across
auctions, implying a limited number of truly homogeneous auctions.

Within our novel asymptotic framework, we introduce new inference methods for
the winner’s expected utility, the seller’s expected revenue, and the
valuation distribution’s tail behavior. We show that the latter can be used to
test the regularity conditions commonly assumed in auction literature. Our
data requirements are minimal; our methods rely on observing transaction
prices from a finite number of auctions. We do not need to observe multiple
bids or the number of participating bidders in these auctions.

Our methodology characterizes the limiting behavior of transaction prices as
the number of bidders increases. Unlike the traditional asymptotic framework
with a growing number of auctions, the transaction price data does not allow
us to fully identify the valuation distribution. However, it can reveal the
distribution’s tail properties, which are sufficient for conducting inference
on practical objects of interest, such as the winner’s expected utility and
the seller’s expected revenue. Moreover, the tail behavior allows us to
evaluate whether the valuation distribution has bounded support and positive
density at its highest value, which is a common regularity in the auction
literature. (e.g., Maskin and Riley, 1984; Guerre et al., 2000; Guerre and
Luo, 2022). Within our asymptotic framework, our inference methods are shown
to control size and exhibit desirable power properties. Specifically, our
confidence intervals are shown to minimize expected length and our hypothesis
tests are shown to maximize weighted average power. Finally, Monte Carlo
simulations confirm the accuracy of our methodology in finite samples, and a
study of Hong Kong car license plate auctions illustrates its empirical
relevance.

Our assumptions are generally mild. We require a large number of bidders and
that the valuation distribution falls within the domain of attraction of the
extreme value (EV) distribution, a condition satisfied by commonly used
distributions like Pareto, Student-t, Gaussian, and uniform distributions (see
Section 2 for details). In this setup, we can conduct inference using
transaction prices from just three or more auctions. Furthermore, the auctions
do not need to have the same number of bidders (which may remain unknown), but
they should diverge at the same rate.

The asymptotic framework with many bidders has been extensively employed in
the economic theory literature (e.g., Hong and Shum, 2004; Virág, 2013; Di
Tillio et al., 2021) but less so in econometrics. Hong et al. (2014) studies
the asymptotic distribution of the transaction price in a clock model of a
multi-unit, oral, ascending-price auction as the numbers of bidders and units
increase. Menzel and Morganti (2013) shows that the nonparametric estimator of
the valuation distribution may become irregular and perform poorly when the
number of bidders increases. This issue does not affect our method, which
relies on the EV approximation. In comparison, much of the existing
econometric literature has focused on identifying and estimating the valuation
distribution. In cases with a small number of bidders, like timber auctions,
identifying the valuation distribution is the key to understanding the
relevant auction features. The seminal work by Athey and Haile (2002) derives
general results about identifying the valuation distribution from the
distribution of bids. Haile and Tamer (2003) investigates English auctions and
establishes bounds on the valuation distribution and other objects of interest
with minimal structural assumptions. Chesher and Rosen (2017) extends these
bounds to the non-IPV setup. Aradillas-Lopez et al. (2013) nonparametrically
identifies bounds on seller profit and bidder surplus while accounting for
variations in the number of bidders across auctions. Brendstrup and Paarsch
(2006) and Komarova (2013) derive nonparametric identification of the
valuation distribution with the transaction price and the winner’s identity.
Brendstrup and Paarsch (2007) investigates multi-object English auctions and
establishes semiparametric identification using winning bids under the
Archimedean copula assumption.

There is a vast literature that considers inference in the more traditional
asymptotic framework in which the number of auctions diverges to infinity. We
now briefly highlight some of its recent contributions. For example, Li (2005)
studies first-price auctions with entry and binding reservation prices and
estimates the valuation distribution with the observed bids and the number of
actual bidders. Without knowing the number of bidders, An et al. (2010)
proposes using a proxy of the number of bidders and an instrument variable.
Kim and Lee (2014), Song (2015), Mbakop (2017), and Freyberger and Larsen
(2022) construct identification of the valuation distribution with two or more
order statistics of bids. Shneyerov and Wong (2011) derive nonparametric
identification of model primitives based on finitely many groups of bidders.
Recently, Luo and Xiao (2023) derives identification results with two
consecutive order statistics and an instrument or three consecutive ones. All
these methods of identification, estimation, and inference are based on the
traditional asymptotic framework with many auctions and multiple bids from
each auction. We refer to Hickman et al. (2012) and Gentry et al. (2018) for
recent surveys.

As already mentioned, our paper provides inference based only on transaction
prices. This aspect of our paper resembles the recent work by Guerre and Luo
(2022). However, there are considerable differences between this paper and our
contribution. In particular, Guerre and Luo (2022) considers first-price
auctions in which the number of bidders in each auction is random and has
finite support. In this context, they establish that the winning bid is
increasing in the number of bidders and, hence, the density of the winning
bids exhibits discontinuities as the number of bidders changes. Our framework
has many differences with Guerre and Luo (2022). First, their identification
strategy relies on the assumption that the number of bidders is finite, while
we specialize in the case when this amount diverges. Thus, the contributions
are designed for different empirical environments. Second, their argument does
not extend to second-price auctions, while ours does. Third, their method
relies on observing a large number of auctions, while ours can deal with
applications with a finite number of them, as long as this number exceeds
three. Finally, their paper delivers identification analysis, while ours
focuses on inference.

The rest of the paper is organized as follows. Section 2 establishes the new
asymptotic framework with many bidders and discusses its relationship with
extreme value (EV) theory. Section 3 focuses on second-price auctions and
introduces our new inference methods. We begin by outlining the auction format
and deriving the asymptotic distribution of transaction prices within the new
framework. Subsequently, Section 3.1 presents confidence intervals for the
winner’s expected utility, Section 3.2 introduces confidence intervals for the
seller’s expected revenue, and Section 3.3 outlines hypothesis tests for the
tail index. Section 4 offers analogous results for first-price auctions.
Section 5 provides the Monte Carlo simulation results, and Section 6 presents
our empirical application. Finally, Section 7 concludes. The paper’s appendix
provides all proofs, auxiliary results, and computational details.

##  2 Asymptotic framework with many bidders

We consider inference in sealed-bid auctions for a single object, where the
data consist of transaction prices from $n\geq 3$ independent auction
realizations, denoted as $\\{P_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,n\\}$. Importantly, our framework does not require $n$ to
diverge to infinity.

For each auction $j=1,\dots,n$, the setup is as follows. There is a single
object for sale, and $K_{j}$ potential buyers are bidding for it. These
bidders have independent private values $\\{V_{i,j}\mathrel{\mathop{\mathchar
58\relax}}i=1,\dots,K_{j}\\}$ distributed according to a common cumulative
distribution function (CDF) $F_{V}$ with support on $[v_{L},v_{H}]$, with
$0\leq v_{L}<v_{H}$, where $v_{H}=\infty$ is allowed. We assume that $F_{V}$
strictly increases on its support and admits a continuous probability density
function (PDF) $f_{V}=F_{V}^{\prime}$. Bidders are assumed to be risk neutral
and maximize expected profits without facing any liquidity or budget
constraints. $F_{V}$ and $K_{j}$ are common knowledge to all bidders, but
unknown to the researcher. Our inference methods rely on the asymptotics with
diverging numbers of bidders $K_{j}$ and a finite number of auctions $n$. To
this end, we assume that $K\equiv\min\\{K_{1},\dots,K_{n}\\}\to\infty$ and
$K_{i}/K\to 1$ for each $i=1,\dots,n$.111If $K_{j}$ varies across auctions,
then these are not identically distributed. That is, we assume all $n$
auctions have approximately the same large number of bidders.222Allowing for
an unknown number of bidders and $K_{j}/K\not\to 1$ for some $j=1,\dots,n$
significantly complicates our inference problem. In particular, this would
require observing more than the transaction price in each auction,
contradicting the core premise of this paper.

We make an additional mild assumption about the valuation distribution
$F_{V}$. We assume that it is in the domain of attraction of the EV
distribution $G_{\xi}$, where $\xi$ denotes the tail index. Formally, this
means that there is a sequence of normalizing constants
$\\{(a_{K},b_{K})\in\mathbb{R}_{++}\times\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}K\in\mathbb{N}\\}$ such that, for all $x$ that is a continuity point
of $G_{\xi}$,

$\lim_{K\to\infty}~{}(F_{V}(a_{K}x+b_{K}))^{K}~{}~{}=~{}~{}G_{\xi}(x).$ (2.1)

See de Haan and Ferreira (2006, Chapter 1) or David and Nagaraja (2004,
Chapter 10) for recent expositions on this topic, including sufficient
conditions for (2.1). Under the condition in (2.1), standard asymptotic
results imply that $G_{\xi}$ belongs to one of three types: Weibull (if
$\xi<0$), Gumbel (if $\xi=0$), or Fréchet (if $\xi>0$). We can unify these
distributions into the generalized EV distribution, with the following CDF:

$G_{\xi}(x)~{}=~{}\left\\{\begin{array}[]{ll}\exp(-(1+\xi x)^{-1/\xi})I(1+\xi
x>0)&\text{ if }\xi>0,\\\ \exp(-\exp(-x))&\text{ if }\xi=0,\\\ \exp(-(1+\xi
x)^{-1/\xi})I(1+\xi x>0)+I(1+\xi x\leq 0)&\text{ if }\xi<0,\end{array}\right.$
(2.2)

See de Haan and Ferreira (2006, Theorem 1.1.3) or David and Nagaraja (2004,
Theorem 10.5.1).

Condition (2.1) is a very mild restriction, as is satisfied by commonly used
valuation distributions $F_{V}$. First, the case with $\xi>0$ covers
distributions with unbounded support (i.e., $v_{H}=\infty$) and polynomial
decaying (i.e., “heavy”) right tail. In this case, moments of order less than
$1/\xi$ exist and moments of order greater than $1/\xi$ do not (see de Haan
and Ferreira (2006, page 176)). Then, the restriction to $\xi\leq 1/2$ implies
that $F_{V}$ has finite second moments. Examples include Pareto, Student-t,
and F distributions. Second, the case with $\xi=0$ encompasses distributions
with unbounded support (i.e., $v_{H}=\infty$) but with exponential decaying
(i.e., “light”) right tail and bounded moments of any order. Examples include
normal and log-normal distributions. Finally, the case $\xi<0$ covers
distributions with bounded support (i.e., $v_{H}<\infty$), such as Beta,
Uniform, and triangular distributions.

For the purpose of this paper, the significance of (2.1) is that it allows us
to characterize the joint distribution of the ordered valuations for all
auctions as the number of bidders diverges. We now introduce the relevant
notation to this end. For each auction $j=1,\dots,n$, let
$\\{V_{(i),j}\mathrel{\mathop{\mathchar 58\relax}}i=1,\dots,K_{j}\\}$ denote
the order statistics of $\\{V_{i,j}\mathrel{\mathop{\mathchar
58\relax}}i=1,\dots,K_{j}\\}$ in decreasing order, i.e., $V_{(1),j}\geq
V_{(2),j}\geq\dots\geq V_{(K_{j}),j}$. Lemma 2.1 provides the joint
distribution of the extreme order statistics for all auctions.

######  Lemma 2.1.

Assume (2.1) holds. For any $n\in\mathbb{N}$ and any $d\in\mathbb{N}$, and as
$K\to\infty$,

$\displaystyle\Big{\\{}\Big{(}\tfrac{V_{(1),j}-b_{K}}{a_{K}},~{}\tfrac{V_{(2),j}-b_{K}}{a_{K}},~{}\dots~{},~{}\tfrac{V_{(d),j}-b_{K}}{a_{K}}\Big{)}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,n\Big{\\}}~{}\overset{d}{\to}~{}$
$\displaystyle\Big{\\{}\Big{(}H_{\xi}(E_{1,j}),~{}H_{\xi}(E_{1,j}+E_{2,j}),~{}\dots~{},~{}H_{\xi}\big{(}\sum\nolimits_{s=1}^{d}E_{s,j}\big{)}\Big{)}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,n\Big{\\}},$ (2.3)

where
$\\{(a_{K},b_{K})\in\mathbb{R}_{++}\times\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}K\in\mathbb{N}\\}$ are the normalizing constants in (2.1),
$\\{E_{s,j}\mathrel{\mathop{\mathchar 58\relax}}s=1,\dots,d,~{}j=1,\dots,n\\}$
are i.i.d. standard exponential random variables, and

$H_{\xi}(x)~{}\equiv~{}\bigg{\\{}\begin{array}[]{cc}({x}^{-\xi}-1)/{\xi}&\text{
if }\xi\neq 0,\\\ -\ln(x)&\text{ if }\xi=0.\end{array}$ (2.4)

Lemma 2.1 characterizes the asymptotic distribution of the largest order
statistics. In this paper, we consider first-price and second-price auction
formats, which involve only $V_{(1),j}$ and $V_{(2),j}$, respectively. We
assume symmetric equilibrium bidding, enabling us to establish a relationship
between private valuations, equilibrium bids, and transaction prices ${\bf
P}\equiv\\{P_{j}\mathrel{\mathop{\mathchar 58\relax}}j=1,\dots,n\\}$. As a
corollary, we can completely describe the asymptotic distribution of
transaction prices in terms of the tail index $\xi$. This, in turn, allows us
to perform inference of several objects of economic interest solely based on
the transaction prices.

##  3 Second-price auctions

We begin our analysis with second-price sealed-bid auctions, in which the
highest bidder wins the object and pays the second-highest bid. Since we
consider a private value framework, second-price auctions are equivalent in a
weak sense to open ascending price (or English) auctions (see Krishna (2009,
page 4)). By standard arguments (e.g., Krishna (2009, Proposition 2.1)), the
weakly dominant strategy for a bidder with valuation $v$ in auction
$j=1,\dots,n$ is

$\beta_{j}(v)~{}=~{}v.$ (3.1)

Thus, the observed transaction price in auction $j$ equals the second-highest
bid, i.e.,

$P_{j}~{}=~{}V_{(2),j}.$ (3.2)

By Lemma 2.1 and (3.2), we conclude that as $K\to\infty$,

$\Big{\\{}\tfrac{P_{j}-b_{K}}{a_{K}}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,n\Big{\\}}~{}\overset{d}{\to}~{}\\{Z_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,n\\},$ (3.3)

where $\\{Z_{j}\mathrel{\mathop{\mathchar 58\relax}}j=1,\dots,n\\}$ is i.i.d.
with $Z_{j}\equiv H_{\xi}(E_{1,j}+E_{2,j})$ for each $j=1,\dots,n$, and
$\\{(a_{K},b_{K})\in\mathbb{R}_{++}\times\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}K\in\mathbb{N}\\}$, $\\{(E_{1,j},E_{2,j})\mathrel{\mathop{\mathchar
58\relax}}~{}j=1,\dots,n\\}$, and $H_{\xi}$ are as in Lemma 2.1.

If the constants
$\\{(a_{K},b_{K})\in\mathbb{R}_{++}\times\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}K\in\mathbb{N}\\}$ were known, we could use (3.3) to perform
inference on functions of the EV index $\xi$. Unfortunately, these constants
are unknown and depend implicitly on the underlying distribution of
valuations. To sidestep this issue, we sort the transaction prices across
auctions (i.e., $P_{(1)}\leq P_{(2)}\leq\dots\leq P_{(n)}$), and consider the
sorted and self-normalized prices: for $j=1,\dots,N\equiv n-2\geq 1$,

$\tilde{P}_{j}~{}\equiv~{}\Bigg{\\{}\begin{array}[]{cc}\tfrac{P_{(j+1)}-P_{(1)}}{P_{(n)}-P_{(1)}}&\text{
if }P_{(n)}>P_{(1)}\\\ 0&\text{ if }P_{(n)}=P_{(1)},\end{array}$ (3.4)

and let $\mathbf{\tilde{P}}=\\{\tilde{P}_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,N\\}\in\Sigma\equiv\\{h\in[0,1]^{N}\mathrel{\mathop{\mathchar
58\relax}}0\leq h_{1}\leq\dots\leq h_{N}\leq 1\\}$. The following result
characterizes the asymptotic distribution of $\mathbf{\tilde{P}}$ as
$K\to\infty$.

######  Lemma 3.1.

Assume (2.1) holds. For any $N\in\mathbb{N}$, and as $K\to\infty$,

$\mathbf{\tilde{P}}=\\{\tilde{P}_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,N\\}~{}\overset{d}{\to}~{}\mathbf{\tilde{Z}}=\\{\tilde{Z}_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,N\\},$ (3.5)

where the joint density of $\mathbf{\tilde{Z}}$ is

$\displaystyle
f_{\mathbf{\tilde{Z}}|\xi}(z_{1},\dots,z_{N})~{}\equiv~{}1[0\leq
z_{1}\leq\dots\leq z_{N}\leq 1]~{}(N+2)!~{}\Gamma(2(N+2))\times$
$\displaystyle~{}\left\\{\begin{array}[]{cc}\int_{0}^{-1/\xi}s^{N}\exp\bigg{(}\begin{array}[]{c}-2(N+2)\ln(\sum_{j=1}^{N}(1+z_{j}\xi
s)^{-1/\xi}+(1+\xi s)^{-1/\xi})\\\ -(1+2/\xi)(\sum_{j=1}^{N}\ln(1+z_{j}\xi
s)+\ln(1+\xi s))\end{array}\bigg{)}ds&\text{if }\xi<0,\\\
\int_{0}^{\infty}s^{N}\exp\bigg{(}\begin{array}[]{c}-2(N+2)\ln(\sum_{j=1}^{N}\exp(-z_{j}s)+\exp(-s))\\\
-2s(\sum_{j=1}^{N}z_{j}+1)\end{array}\bigg{)}ds&\text{if }\xi=0,\\\
\int_{0}^{\infty}s^{N}\exp\bigg{(}\begin{array}[]{c}-2(N+2)\ln(\sum_{j=1}^{N}(1+z_{j}\xi
s)^{-1/\xi}+(1+\xi s)^{-1/\xi})\\\ -(1+2/\xi)(\sum_{j=1}^{N}\ln(1+z_{j}\xi
s)+\ln(1+\xi s))\end{array}\bigg{)}ds&\text{if }\xi>0,\end{array}\right.$
(3.15)

and $\Gamma$ is the standard Gamma function.

Lemma 3.1 reveals that the asymptotic distribution of $\mathbf{\tilde{P}}$ is
informative about the tail index $\xi$. In the next subsections, we show how
to use this information to conduct asymptotically valid inference on the tail
index $\xi$ and several other important features of these auctions, such as
the winner’s expected utility and the seller’s expected revenue.

###  3.1 Inference about the winner’s expected utility

Our objective is to conduct inference on the average of the winner’s expected
utility based on the transaction prices. Since each bidder bids their own
valuation, the auction is won by the highest bidder, and the transaction price
is equal to the second-highest bid, we conclude that the winner’s expected
utility in auction $j=1,\dots,n$ is
$E[V_{(1),j}-P_{j}]=E[V_{(1),j}-V_{(2),j}]$, whose average is

$\mu_{K}~{}=~{}\frac{1}{n}\sum_{j=1}^{n}E[V_{(1),j}-V_{(2),j}].$ (3.16)

Given transaction prices $\mathbf{P}$, we consider a confidence interval (CI)
for $\mu_{K}$ given by

$U(\mathbf{P})~{}=~{}(P_{(n)}-P_{(1)})\times\tilde{U}(\mathbf{\tilde{P}}),$
(3.17)

where $\mathbf{\tilde{P}}\in\Sigma$ are the sorted and self-normalized
transaction prices in (3.4), and $\tilde{U}\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\mathcal{P}(\mathbb{R})$ is a CI defined on
$\mathbf{\tilde{P}}$. By (3.17), the CI $U(\mathbf{P})$ is invariant to the
sorting and translation of $\mathbf{P}$, and equivariant to their scale.

The remainder of the section denotes $Z_{1,j}=H_{\xi}(E_{1,j})$,
$Z_{2,j}=H_{\xi}(E_{1,j}+E_{2,j})$,
$\\{(E_{1,j},E_{2,j})\mathrel{\mathop{\mathchar 58\relax}}j=1,\dots,n\\}$ are
i.i.d. standard exponential random variables and $H_{\xi}$ is as in (2.4). We
also use $\mathbf{Z}=\\{Z_{2,j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,n\\}$, $Z_{(n)}=\max\\{Z_{2,j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,n\\}$, $Z_{(1)}=\min\\{Z_{2,j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,n\\}$, for $j=1,\ldots,N=n-2$,

$\tilde{Z}_{j}~{}=~{}\bigg{\\{}\begin{tabular}[]{ll}$\tfrac{Z_{(j+1)}-Z_{(1)}}{Z_{(n)}-Z_{(1)}}$&if
$Z_{(n)}>Z_{(1)},$\\\ $0$&if $Z_{(n)}=Z_{(1)},$\end{tabular}$ (3.18)

and $\mathbf{\tilde{Z}}=\\{\tilde{Z}_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,N\\}\in\Sigma$. Finally, let $Y_{\mu}\equiv
E[Z_{1,1}-Z_{2,1}]/(Z_{(n)}-Z_{(1)})$ and
$\kappa_{\xi}(\mathbf{\tilde{Z}})=E[Z_{(n)}-Z_{(1)}|\mathbf{\tilde{Z}}]$. The
distributions of these random variables are fully characterized by the tail
index $\xi$. In particular, Lemma A.4 shows
$E[Z_{1,1}-Z_{2,1}]=\Gamma(1-\xi)$, where $\Gamma$ is the standard Gamma
function. For the remainder of this section, we will use $P_{\xi}$ and
$E_{\xi}$ to refer to the probability and expectation associated with this
distribution. The following result describes the asymptotic properties of the
CI in (3.17) as $K\to\infty$.

######  Theorem 3.1.

Assume (2.1) holds, and that for some $\varepsilon>0$ with
$(1+\varepsilon)\xi<1$, $E[|V_{i,j}|^{1+\varepsilon}]<\infty$ for all
$i=1\dots,K_{j}$ in auction $j=1,\dots,N$. Finally, assume that the CI for
$\mu_{K}$, $U(\mathbf{P})$, is as in (3.17) with
$\tilde{U}\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\mathcal{P}(\mathbb{R})$ that satisfies the following
conditions:

  1. (a)

$P_{\xi}(\\{Y_{\mu},\mathbf{\tilde{Z}}\\}\in\partial\\{(y,h)\in\mathbb{R}\times\Sigma\mathrel{\mathop{\mathchar
58\relax}}y\in\tilde{U}(h)\\})=0$, where $\partial A$ denotes the boundary of
$A$.

  2. (b)

$\lg(\tilde{U}(h))<\infty$ for any $h\in\Sigma$, where $\lg(A)$ denotes the
length of $A$ (i.e., $\lg(A)\equiv\int\mathbf{1}[y\in A]dy$).

  3. (c)

For any sequence $\\{h_{\ell}\in\Sigma\\}_{\ell\in\mathbb{N}}$ with
$h_{\ell}\to h\in\Sigma$, $\lg(\tilde{U}(h_{\ell}))\to\lg(\tilde{U}(h))$.

Then, as $K\to\infty$,

  1. 1.

$P(\mu_{K}\in
U(\mathbf{P}))~{}\to~{}P_{\xi}(Y_{\mu}\in\tilde{U}(\mathbf{\tilde{Z}}))$,

  2. 2.

$E[\lg(U(\mathbf{P}))]/a_{K}~{}\to~{}E_{\xi}[\kappa_{\xi}(\mathbf{\tilde{Z}})\lg(\tilde{U}(\mathbf{\tilde{Z}}))]$.

Suppose that we consider the class of CIs for $\mu_{K}$ given by (3.17) and
conditions (a)-(c) in Theorem 3.1. The finite sample properties of these CIs
are unknown. If the number of bidders is large, it is natural to rely on the
asymptotic behavior derived in Theorem 3.1 to choose our CI. First, we can
guarantee asymptotic validity by imposing

$P_{\xi}(Y_{\mu}\in\tilde{U}(\mathbf{\tilde{Z}}))~{}\geq~{}1-\alpha~{}~{}~{}\text{
for all }\xi\in\Xi.$ (3.19)

Second, we can seek improvements in statistical power by choosing a CI that
has a small asymptotic expected length (scaled by $a_{K}>0$). Since the tail
index $\xi$ is unknown, we focus on the CI’s asymptotic weighted length, given
by

$\int_{\xi\in\Xi}E_{\xi}[\kappa_{\xi}(\tilde{\mathbf{{Z}}})\lg(\tilde{U}(\tilde{\mathbf{{Z}}})))]dW(\xi),$
(3.20)

where $W$ is a user-defined weight function. We can combine both objectives by
choosing the CI for $\mu_{K}$ that minimizes the asymptotic weighted length in
(3.20) subject to asymptotic validity condition in (3.19).

Formally, let $\mathbb{U}$ denote the collection of CIs that satisfy
conditions (a)-(c) in Theorem 3.1. Then, we propose choosing $\tilde{U}$ in
(3.17) as the solution to the following problem:

$\underset{\tilde{U}\in\mathbb{U}}{\arg\min}\int_{\xi\in\Xi}E_{\xi}[\kappa_{\xi}(\tilde{\mathbf{{Z}}})\lg(\tilde{U}(\tilde{\mathbf{{Z}}})))]dW(\xi)~{}~{}\text{
s.t.}~{}~{}P_{\xi}(Y_{\mu}\in\tilde{U}(\mathbf{\tilde{Z}}))~{}\geq~{}1-\alpha~{}~{}\text{
for all }\xi\in\Xi.$ (3.21)

Following Müller and Wang (2017), we write down (3.21) in its Lagrangian form:

$\min_{\tilde{U}\in\mathbb{U}}\int_{\xi\in\Xi}E_{\xi}[\kappa_{\xi}(\tilde{\mathbf{{Z}}})\lg(\tilde{U}(\tilde{\mathbf{{Z}}})))]dW(\xi)+\int_{\xi\in\Xi}P_{\xi}(Y_{\mu}\in\tilde{U}(\mathbf{\tilde{Z}}))d\Lambda(\xi),$
(3.22)

where the non-negative measure $\Lambda$ denotes the Lagrangian weights chosen
to guarantee the asymptotic size constraint in (3.21). If we ignore the
constraints in $\mathbb{U}$, the solution to (3.22) is given by the following
set: for every $h\in\Sigma$,

$\tilde{U}(h)~{}=~{}\bigg{\\{}y\in\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}\int_{\xi\in\Xi}\kappa_{\xi}(h)f_{\tilde{\mathbf{Z}}|\xi}(h)dW(\xi)\leq\int_{\xi\in\Xi}f_{(Y_{\mu},\tilde{\mathbf{Z}})|\xi}(y,h)d\Lambda(\xi)\bigg{\\}},$
(3.23)

where for every $(y,h)\in\mathbb{R}\times\Sigma$,
$\kappa_{\xi}(h)f_{\tilde{\mathbf{Z}}|\xi}(h)$ and
$f_{(Y_{\mu},\tilde{\mathbf{Z}})|\xi}(y,h)$ are given in Section A.3, and the
integrals in (3.23) can be numerically calculated by Gaussian quadrature. We
can verify numerically that $\tilde{U}(h)$ in (3.23) takes the form of an
interval whose length varies continuously with $h\in\Sigma$.333Given the
intricate nature of the densities, we were unable to prove this result
analytically. Under these conditions, Lemma A.3 shows that $\tilde{U}(h)$ in
(3.23) belongs to $\mathbb{U}$ and, therefore, $\tilde{U}(h)$ in (3.23) solves
(3.22).

By combining (3.17), (3.23), and the arguments in the previous paragraph, we
propose:

$U(\mathbf{P})~{}=~{}(P_{(n)}-P_{(1)})\times\bigg{\\{}y\in\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}\int_{\xi\in\Xi}\kappa_{\xi}(\mathbf{\tilde{P}})f_{\tilde{\mathbf{Z}}|\xi}(\mathbf{\tilde{P}})dW(\xi)\leq\int_{\xi\in\Xi}f_{(Y_{\mu},\tilde{\mathbf{Z}})|\xi}(y,\mathbf{\tilde{P}})d\Lambda(\xi)\bigg{\\}}.$
(3.24)

Under the conditions in the previous paragraph, Theorem 3.1 implies that
(3.24) belongs to the class of asymptotically valid CIs and minimizes the
asymptotic expected length (scaled by $a_{K}>0$) within this class.

To calculate (3.24), the only remaining challenge is to find appropriate
Lagrangian weights $\Lambda$ that ensure asymptotic validity in the limiting
problem, as described in (3.19). We tackle this challenge using a numerical
approach developed by Elliott et al. (2015). It is relevant to emphasize that
these Lagrangian weights depend solely on the value of $n$ and, as a result,
they only need to be computed once. For more information on calculating these
weights, please refer to Section A.3.

###  3.2 Inference about the seller’s expected revenue

We now conduct inference on the average of the seller’s expected revenue based
on transaction prices. Since bidders bid their valuation and the transaction
price is the second-highest bid, we get that the seller’s expected revenue in
auction $j=1,\dots,n$ is $E[P_{j}]=E[V_{(2),j}]$, whose average is

$\pi_{K}~{}=~{}\frac{1}{n}\sum_{j=1}^{n}E[V_{(2),j}].$ (3.25)

Given transaction prices $\mathbf{P}$, we consider a CI for $\pi_{K}$ given by

$U(\mathbf{P})~{}\equiv~{}(P_{(n)}-P_{(1)})\times\tilde{U}(\mathbf{\tilde{P}})+P_{(1)},$
(3.26)

where $\mathbf{\tilde{P}}\in\Sigma$ are the sorted and self-normalized
transaction prices in (3.4), and $\tilde{U}\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\mathcal{P}(\mathbb{R})$ is a CI defined on
$\mathbf{\tilde{P}}$. By (3.26), the CI $U(\mathbf{P})$ is invariant to the
sorting of $\mathbf{P}$, and equivariant to their location and scale.

The remainder of this section denotes $Z_{2,j}=H_{\xi}(E_{1,j}+E_{2,j})$,
$\\{(E_{1,j},E_{2,j})\mathrel{\mathop{\mathchar 58\relax}}j=1,\dots,n\\}$ are
i.i.d. standard exponential random variables and $H_{\xi}$ is as in (2.4). We
also use $\mathbf{Z}=\\{Z_{2,j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,n\\}$, $Z_{(n)}=\max\\{Z_{2,j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,n\\}$, $Z_{(1)}=\min\\{Z_{2,j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,n\\}$, for $j=1,\ldots,N=n-2$,

$\tilde{Z}_{j}=\bigg{\\{}\begin{tabular}[]{ll}$\tfrac{Z_{(j+1)}-Z_{(1)}}{Z_{(n)}-Z_{(1)}}$&if
$Z_{(n)}>Z_{(1)},$\\\ $0$&if $Z_{(n)}=Z_{(1)},$\end{tabular}$

and $\mathbf{\tilde{Z}}=\\{\tilde{Z}_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,N\\}\in\Sigma$. Finally, let $Y_{\pi}\equiv
E[Z_{2,1}]/(Z_{(n)}-Z_{(1)})$ and
$\kappa_{\xi}(\mathbf{\tilde{Z}})=E[Z_{(n)}-Z_{(1)}|\mathbf{\tilde{Z}}]$. As
in the previous section, the distributions of these random variables are fully
characterized by its tail index $\xi$. In particular, Lemma A.5 shows that
$E[Z_{2,1}]=(\Gamma(2-\xi)-1)/\xi$ if $\xi\neq 0$, and $-1+\bar{\gamma}$ if
$\xi=0$, where $\Gamma$ is the standard Gamma function and
$\bar{\gamma}\approx 0.577$ is the Euler’s constant. For the remainder of this
section, we will use $P_{\xi}$ and $E_{\xi}$ to refer to the probability and
expectation associated with this distribution. The next result provides the
asymptotic properties of the CI in (3.26) as $K\to\infty$.

######  Theorem 3.2.

Assume (2.1) holds, and that for some $\varepsilon>0$ with
$(1+\varepsilon)\xi<1$, $E[|V_{i,j}|^{1+\varepsilon}]<\infty$ for all
$i=1\dots,K_{j}$ in auction $j=1,\dots,N$. Finally, assume that the CI for
$\pi_{K}$, $U(\mathbf{P})$, is as in (3.26) with
$\tilde{U}\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\mathcal{P}(\mathbb{R})$ that satisfies the following
conditions:

  1. (a)

$P_{\xi}(\\{Y_{\pi},\mathbf{\tilde{Z}}\\}\in\partial\\{(y,h)\in\mathbb{R}\times\Sigma\mathrel{\mathop{\mathchar
58\relax}}y\in\tilde{U}(h)\\})=0$, where $\partial A$ denotes the boundary of
$A$.

  2. (b)

$\lg(\tilde{U}(h))<\infty$ for any $h\in\Sigma$, where $\lg(A)$ denotes the
length of $A$.

  3. (c)

For any sequence $\\{h_{\ell}\in\Sigma\\}_{\ell\in\mathbb{N}}$ with
$h_{\ell}\to h\in\Sigma$, $\lg(\tilde{U}(h_{\ell}))\to\lg(\tilde{U}(h))$.

Then, as $K\to\infty$,

  1. 1.

$P(\pi_{K}\in
U(\mathbf{P}))~{}\to~{}P_{\xi}(Y_{\pi}\in\tilde{U}(\mathbf{\tilde{Z}}))$,

  2. 2.

$E[\lg(U(\mathbf{P}))]/a_{K}~{}\to~{}E_{\xi}[\kappa_{\xi}(\mathbf{\tilde{Z}})\lg(\tilde{U}(\mathbf{\tilde{Z}}))]$.

Following the ideas in Section 3.2, we choose our CI based on the asymptotic
behavior in Theorem 3.2. In particular, we propose to choose $\tilde{U}$ in
(3.26) to minimize the asymptotic weighted length of the CI subject to
asymptotic validity, given by

$\underset{\tilde{U}\in\mathbb{U}}{\arg\min}\int_{\xi\in\Xi}E_{\xi}[\kappa_{\xi}(\tilde{\mathbf{{Z}}})\lg(\tilde{U}(\tilde{\mathbf{{Z}}})))]dW(\xi)~{}~{}~{}\text{
s.t.}~{}~{}~{}P_{\xi}(Y_{\pi}\in\tilde{U}(\mathbf{\tilde{Z}}))~{}\geq~{}1-\alpha~{}~{}~{}\text{
for all }\xi\in\Xi,$ (3.27)

where $\mathbb{U}$ denotes the collection of CIs that satisfy the conditions
in Theorem 3.2 and $W$ is a user-defined weight function. The solution to
(3.27) is as follows: for every $h\in\Sigma$,

$\tilde{U}(h)~{}=~{}\bigg{\\{}y\in\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}\int_{\xi\in\Xi}\kappa_{\xi}(h)f_{\tilde{\mathbf{Z}}|\xi}(h)dW(\xi)\leq\int_{\xi\in\Xi}f_{(Y_{\pi},\tilde{\mathbf{Z}})|\xi}(y,h)d\Lambda(\xi)\bigg{\\}},$
(3.28)

where, for every $(y,h)\in\mathbb{R}\times\Sigma$,
$\kappa_{\xi}(h)f_{\tilde{\mathbf{Z}}|\xi}(h)$ and
$f_{(Y_{\pi},\tilde{\mathbf{Z}})|\xi}(y,h)$ are given in Section A.3.

By combining (3.17), (3.23), and the arguments in the previous paragraph, we
propose:

$\displaystyle U(\mathbf{P})~{}=~{}$ $\displaystyle P_{(1)}+(P_{(n)}-P_{(1)})$
$\displaystyle\times$
$\displaystyle\bigg{\\{}y\in\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}\int_{\xi\in\Xi}\kappa_{\xi}(\mathbf{\tilde{P}})f_{\tilde{\mathbf{Z}}|\xi}(\mathbf{\tilde{P}})dW(\xi)\leq\int_{\xi\in\Xi}f_{(Y_{\pi},\tilde{\mathbf{Z}})|\xi}(y,\mathbf{\tilde{P}})d\Lambda(\xi)\bigg{\\}},$
(3.29)

Our derivations establish that (3.29) belongs to the class of asymptotically
valid CIs and minimizes the asymptotic expected length (scaled by $a_{K}>0$)
within this class.

###  3.3 Inference about the tail index

A key parameter in our asymptotic framework is the tail index $\xi\in\Xi$. The
goal of this section is to conduct inference on this parameter. That is, we
are interested in the following hypothesis test:

$H_{0}\mathrel{\mathop{\mathchar
58\relax}}\xi\in\xi_{0}~{}~{}~{}\text{v.s.}~{}~{}~{}H_{1}\mathrel{\mathop{\mathchar
58\relax}}\xi\in\Xi_{1},$ (3.30)

where $\xi_{0}$ is a fixed parameter value in $\Xi$ and
$\Xi_{1}=\Xi\backslash\\{\xi_{0}\\}$.

We divide this section into three subsections. Section 3.3.1 considers the
situation where the alternative hypothesis in (3.30) is simple, while Section
3.3.2 explores the case where this alternative hypothesis is composite.
Finally, Section 3.3.3 applies the methods in previous sections to test the
standard regularity conditions in auction models.

####  3.3.1 Simple alternative hypothesis

This section considers the following inference problem:

$H_{0}\mathrel{\mathop{\mathchar
58\relax}}\xi=\xi_{0}~{}~{}~{}\text{v.s.}~{}~{}~{}H_{1}\mathrel{\mathop{\mathchar
58\relax}}\xi=\xi_{1},$ (3.31)

where $\xi_{0}$ and $\xi_{1}$ are distinct parameter values.

By the Neyman-Pearson Lemma, a natural starting point is the likelihood ratio
test for the sorted and self-normalized version of our data. While the
likelihood ratio test is unknown in finite samples, its limiting distribution
is provided in Lemma 3.1. This idea yields the following testing procedure:

${\varphi}^{\ast}(\mathbf{{Z}})~{}\equiv~{}1\big{[}{f_{\mathbf{\tilde{Z}}|\xi_{1}}(\mathbf{\tilde{Z}})}/{f_{\mathbf{\tilde{Z}}|\xi_{0}}(\mathbf{\tilde{Z}})}>q(\xi_{0},\xi_{1},\alpha)\big{]},$
(3.32)

where the critical value is given by

$q(\xi_{0},\xi_{1},\alpha)~{}\equiv~{}(1-\alpha)\text{-quantile of
}{f_{\mathbf{\tilde{Z}}|\xi_{1}}(\mathbf{\tilde{Z}}_{0})}/{f_{\mathbf{\tilde{Z}}|\xi_{0}}(\mathbf{\tilde{Z}}_{0})}$

and $\mathbf{\tilde{Z}}_{0}$ is distributed according to
$f_{\mathbf{\tilde{Z}}|\xi_{0}}$. The Neyman-Pearson Lemma implies that (3.32)
is the most powerful level-$\alpha$ test in the limiting problem.

Following the guidance of the asymptotic analysis, our candidate for optimal
test follows from replacing in (3.32) the limiting random variable
$\mathbf{\tilde{Z}}$ with its data analog $\mathbf{\tilde{P}}$, i.e.,

$\varphi_{K}^{\ast}(\mathbf{P})~{}\equiv~{}1\big{[}{f_{\mathbf{\tilde{Z}}|\xi_{1}}(\mathbf{\tilde{P}})}/{f_{\mathbf{\tilde{Z}}|\xi_{0}}(\mathbf{\tilde{P}})}>q(\xi_{0},\xi_{1},\alpha)\big{]}.$
(3.33)

By Lemma 3.1 and standard convergence arguments, (3.33) is asymptotically
valid, i.e.,

$\lim_{K\to\infty}E_{\xi_{0}}[\varphi_{K}^{\ast}(\mathbf{P})]~{}\leq~{}\alpha,$
(3.34)

where $E_{\xi}$ denotes the expectation with respect to distribution with tail
index $\xi$. In fact, (3.33) is asymptotically level $\alpha$, i.e., (3.34)
holds with equality. More interestingly, we can leverage Müller (2011, Theorem
1) to establish that (3.33) is efficient in the sense of being the
asymptotically most powerful test in the class of asymptotically valid and
equivariant tests. Formally, for any asymptotically valid test
$\varphi_{K}(\mathbf{P})$ (i.e., (3.34) holds with
$\varphi_{K}^{\ast}(\mathbf{P})$ replaced by $\varphi_{K}(\mathbf{P})$),
$\underset{K\to\infty}{\lim\sup}~{}E_{\xi_{1}}[\varphi_{K}(\mathbf{P})]~{}\leq~{}\underset{K\to\infty}{\lim\sup}~{}E_{\xi_{1}}[\varphi_{K}^{\ast}(\mathbf{{{P}}})].$
We record these conclusions in the following result.

######  Theorem 3.3.

Assume (2.1) holds. In the hypothesis testing problem in (3.31), the test
defined by (3.33) satisfies the following properties:

  1. 1.

It is asymptotically valid and level $\alpha$, i.e., (3.34) holds with
equality.

  2. 2.

It is asymptotically efficient.

####  3.3.2 Composite alternative hypothesis

We now turn our attention to the case in which the alternative hypothesis in
(3.30) is composite. Following the ideas used in Section 3.3.1, we consider
the feasible version of the efficient test in the limiting problem.
Unfortunately, the limiting problem does not lend itself to the usual tools to
develop uniformly most powerful tests.444In particular, the likelihood ratio
statistic is not monotonic, rendering the results in Lehmann and Romano (2005,
Section 3.4) inapplicable. For this reason, we consider tests that maximize
the weighted average power criterion (WAP), following the approach of Wald
(1943); Andrews and Ploberger (1994). To this end, let $W$ denote a user-
defined weight function on $\Xi_{1}$, which the researcher chooses to reflect
the importance attached to the various alternative hypotheses. The weighting
function $W$ effectively transforms the composite alternative into a simple
one, allowing us to focus on the weighted average power:

$\mathrm{WAP}_{K}(\varphi_{K})~{}\equiv~{}\int_{\Xi_{1}}E_{\xi}[\varphi_{K}(\mathbf{\tilde{P}})]dW(\xi).$

As in the previous section, we use the asymptotic behavior to guide the
construction of our hypothesis test. The likelihood ratio test in the limiting
problem applied to its data $\mathbf{\tilde{P}}$ is given by

$\varphi_{K}^{\ast}(\mathbf{P})~{}\equiv~{}1\bigg{[}{\int_{\Xi_{1}}f_{\mathbf{\tilde{Z}}|\xi}(\mathbf{\tilde{P}})dW(\xi)}/{f_{\mathbf{\tilde{Z}}|\xi_{0}}(\mathbf{\tilde{P}})}>q(\xi_{0},W,\alpha)\bigg{]},$
(3.35)

where the critical value is

$q(\xi_{0},W,\alpha)~{}\equiv~{}(1-\alpha)\text{-quantile of
}{\int_{\Xi_{1}}f_{\mathbf{\tilde{Z}}|\xi_{0}}(\mathbf{\tilde{Z}_{0}})dW(\xi)}/{f_{\mathbf{\tilde{Z}}|\xi_{0}}(\mathbf{\tilde{Z}_{0}})},$

and $\mathbf{\tilde{Z}}_{0}$ is distributed according to
$f_{\mathbf{\tilde{Z}}|\xi_{0}}$. By standard asymptotic arguments, we can
establish that (3.35) is asymptotically valid and level $\alpha$, i.e.,

$\lim_{K\to\infty}E_{\xi_{0}}[\varphi_{K}^{\ast}(\mathbf{P})]~{}=~{}\alpha.$
(3.36)

Moreover, (3.35) is efficient in the sense of maximizing the asymptotic
weighted average power criterion in the class of asymptotically valid and
equivariant tests. Formally, for any test $\varphi_{K}(\mathbf{P})$ that is
asymptotically valid (i.e., (3.36) holds with $\varphi_{K}^{\ast}$ replaced by
$\varphi_{K}$), then
$\underset{K\to\infty}{\lim\sup}~{}\mathrm{WAP}_{K}(\varphi_{K})~{}\leq~{}\underset{K\to\infty}{\lim\sup}~{}\mathrm{WAP}_{K}(\varphi_{K}^{\ast}).$
The next result records these conclusions.

######  Theorem 3.4.

Assume (2.1) holds. In the hypothesis testing problem in (3.30), the test
defined by (3.35) satisfies the following properties:

  1. 1.

It is asymptotically valid and level $\alpha$, i.e., (3.36) holds.

  2. 2.

It is asymptotically efficient.

####  3.3.3 Testing the regularity conditions in the auction literature

The auctions literature routinely assumes the regularity condition that
$f_{V}$ is continuous, and that there is a finite maximum valuation (i.e.,
$v_{H}<\infty$) with $f_{V}(v_{H})>0$. For examples of this, see Maskin and
Riley (1984), Guerre et al. (2000), and Guerre and Luo (2022). Within our
asymptotic framework, the next result shows that these regularity conditions
imply that (2.1) holds with $\xi=-1$.

######  Lemma 3.2.

Assume that $f_{V}(v)\to f_{V}(v_{H})>0$ as $v\uparrow v_{H}<\infty$. Then,
(2.1) holds with $\xi=-1$.

In light of Lemma 3.2, we can test the regularity conditions used in the
auction literature via the following hypothesis test:

$H_{0}\mathrel{\mathop{\mathchar 58\relax}}\xi=-1~{}~{}\text{ v.s.
}~{}~{}H_{1}\mathrel{\mathop{\mathchar 58\relax}}\xi\in\Xi_{1},$ (3.37)

where $\Xi_{1}=\Xi\backslash\\{-1\\}$. In the remainder of this section, we
will argue that $\Xi=[-1,0.5]$ is a suitable choice for the parameters space
for $\xi$. Since (3.37) is a special case of (3.30) with $\Xi_{1}=(-1,0.5]$,
we can implement this test using the procedure in (3.35). For concreteness, we
consider uniform weight $W(\xi)=1[\xi\in(-1,0.5]]$. By Theorem 3.4, our
proposed test is asymptotically valid, level-$\alpha$, and efficient.

We now justify choosing $\Xi=[-1,0.5]$ as the parameter space for the test. We
rely on the so-called von Mises’ condition to interpret the different values
of $\xi$. de Haan and Ferreira (2006, Theorem 1.1.8) states this condition and
shows that it is a sufficient condition for (2.1). Under the von Mises’
condition and that $f_{V}^{\prime}$ is bounded, we have three possible cases:

  1. 1.

$f_{V}(v)\to f_{V}(v_{H})>0$ as $v\to v_{H}<\infty$ implies that $\xi=-1$.

  2. 2.

$f_{V}(v)\to f_{V}(v_{H})=0$ as $v\to v_{H}<\infty$ implies that
$\xi\in(-1,0]$.

  3. 3.

$f_{V}(v)\to f_{V}(v_{H})=0$ as $v\to v_{H}=\infty$ implies that $\xi>0$.

A few remarks are in order. First, as expected, Case 1 aligns with the
findings of Lemma 3.2. Second, we note that de Haan and Ferreira (2006,
Theorem 2.1.2) shows that $v_{H}<\infty$ if and only if $\xi\leq 0$. Cases 2
and 3 then follow from derivations in de Haan and Ferreira (2006, page 18) and
Falk et al. (1994, Theorem 2.1.2). Since these three cases are exhaustive, we
deduce that $\xi\geq-1$. Finally, if we also impose that $V$ has second
moments, de Haan and Ferreira (2006, page 176) implies that $\xi\leq 1/2$.
Combining these restrictions, we conclude that $\Xi=[-1,0.5]$ is a suitable
parameter space for $\xi$.

##  4 First-price auctions

We now consider first-price sealed-bid actions, in which the highest bidder
gets the object and pays the highest bid. This type of auction is
strategically equivalent to an open descending price (or Dutch) auction (see
Krishna (2009, page 4)). By standard arguments (e.g., Krishna (2009,
Proposition 2.2)), the symmetric equilibrium strategy for a bidder with
valuation $v$ in an auction with $K_{j}$ participants is to bid
$\beta_{j}(v)\equiv E[\check{V}_{(1),j}|\check{V}_{(1),j}<v]$, where
$\check{V}_{(1),j}$ denotes the highest bid among the remaining $(K_{j}-1)$
participants. Note that

$\displaystyle\beta_{j}(v)~{}\overset{(1)}{=}~{}\tfrac{K_{j}-1}{F_{V}(v)^{K_{j}-1}}\int_{v_{L}}^{v}{uF_{V}(u)^{K_{j}-2}f_{V}(u)}du~{}=~{}v-\tfrac{\int_{v_{L}}^{v}{F_{V}(u)^{K_{j}-1}}du}{F_{V}(v)^{K_{j}-1}},$
(4.1)

where (1) holds by computing $E[\check{V}_{(1),j}|\check{V}_{(1),j}<v]$ using
the fact that the remaining $(K_{j}-1)$ participants have a common valuation
distribution with PDF $f_{V}$. Since $\beta(v)$ is increasing, the auction is
won by the highest-valuation bidder, with a transaction price equal to

$P_{j}~{}=~{}V_{(1),j}-\tfrac{\int_{v_{L}}^{V_{(1),j}}{F_{V}(u)^{K_{j}-1}}du}{F_{V}(V_{(1),j})^{K_{j}-1}}.$
(4.2)

Lemma A.7 uses (4.2) to deduce that

$\Big{\\{}\tfrac{P_{j}-b_{K}}{a_{K}}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,n\Big{\\}}~{}\overset{d}{\to}~{}\\{X_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,n\\},$ (4.3)

where, for each $j=1,\dots,n$,

$X_{j}~{}\equiv~{}H_{\xi}(E_{1,j})-\tfrac{\int_{-\infty}^{H_{\xi}(E_{1,j})}G_{\xi}(h)dh}{G_{\xi}(H_{\xi}(E_{1,j}))},$
(4.4)

with $G_{\xi}$ is as in (2.2), and
$\\{(a_{K},b_{K})\in\mathbb{R}_{++}\times\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}K\in\mathbb{N}\\}$, $\\{E_{1,j}\mathrel{\mathop{\mathchar
58\relax}}~{}j=1,\dots,n\\}$, and $H_{\xi}$ are as specified in Lemma 2.1.

As in Section 3, the statement in (4.3) cannot be directly used for inference
as it requires the unknown normalizing constants
$\\{(a_{K},b_{K})\in\mathbb{R}_{++}\times\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}K\in\mathbb{N}\\}$. Nevertheless, we can reiterate the idea of
considering sorted and self-normalized prices in (3.4); for
$j=1,\dots,N=n-2\geq 1$, let

$\tilde{P}_{j}~{}\equiv~{}\bigg{\\{}\begin{array}[]{cc}\tfrac{P_{(j+1)}-P_{(1)}}{P_{(n)}-P_{(1)}}&\text{
if }P_{(n)}>P_{(1)}\\\ 0&\text{ if }P_{(n)}=P_{(1)},\end{array}$ (4.5)

and let $\mathbf{\tilde{P}}=\\{\tilde{P}_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,N\\}\in\Sigma$. The next result characterizes the
asymptotic distribution of $\mathbf{\tilde{P}}$ as $K\to\infty$.

######  Lemma 4.1.

Assume (2.1) holds. For any $N\in\mathbb{N}$, and as $K\to\infty$,

$\mathbf{\tilde{P}}=\\{\tilde{P}_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,N\\}~{}\overset{d}{\to}~{}\mathbf{\tilde{X}}=\\{\tilde{X}_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,N\\},$ (4.6)

where $\mathbf{\tilde{X}}=\\{\tilde{X}_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,N\\}\in\Sigma$ is obtained as follows: for
$j=1,\dots,N=n-2\geq 1$,

$\tilde{X}_{j}~{}\equiv~{}\bigg{\\{}\begin{array}[]{cc}\tfrac{X_{(j+1)}-X_{(1)}}{X_{(n)}-X_{(1)}}&\text{
if }X_{(n)}>X_{(1)}\\\ 0&\text{ if }X_{(n)}=X_{(1)},\end{array}$ (4.7)

with $\\{X_{j}\mathrel{\mathop{\mathchar 58\relax}}j=1,\dots,n\\}$ is i.i.d.
according to (4.4).

Given that the random variable in (4.4) is informative about the tail EV index
$\xi$, Lemma 4.1 implicitly reveals that the asymptotic distribution of
$\mathbf{\tilde{P}}$ can be used to conduct inference on functions of $\xi$.
From this point onward, the remainder of this section is analogous to that of
Section 3. The main difference is that the explicit PDF of the asymptotic
distribution of $\mathbf{\tilde{P}}$ in Lemma 3.1 is replaced by the implicit
distribution in Lemma 4.1.

###  4.1 Inference about the winner’s expected utility

Our goal is to conduct inference on the average of the winner’s expected
utility using the transaction prices. Since bidders act according to (4.1),
the auction is won by the bidder with the highest valuation. From here, we
conclude that the winner’s expected utility in auction $j=1,\dots,n$ is
$E[V_{(1),j}-P_{j}]$, whose average is

$\mu_{K}~{}{=}~{}\frac{1}{n}\sum_{j=1}^{n}E\Big{[}\tfrac{\int_{-\infty}^{V_{(1),j}}{F_{V}(u)^{K_{j}-1}}du}{F_{V}(V_{(1),j})^{K_{j}-1}}\Big{]}.$
(4.8)

By the Revenue Equivalence Theorem (e.g., Krishna (2009, Section 3)),
$\mu_{K}$ coincides with the average of the winner’s expected utility in
second-price auctions.

The construction of the CI for $\mu_{K}$ closely follows the arguments and
derivations presented in Section 3.1. In particular, we propose a CI for
$\mu_{K}$ given by

$U(\mathbf{P})~{}=~{}(P_{(n)}-P_{(1)})\times\bigg{\\{}y\in\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}\int_{\xi\in\Xi}\kappa_{\xi}(\mathbf{\tilde{P}})f_{\tilde{\mathbf{X}}|\xi}(\mathbf{\tilde{P}})dW(\xi)\leq\int_{\xi\in\Xi}f_{(Y_{\mu},\tilde{\mathbf{X}})|\xi}(y,\mathbf{\tilde{P}})d\Lambda(\xi)\bigg{\\}},$
(4.9)

where $\mathbf{\tilde{P}}=\\{\tilde{P}_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,N\\}$ with $\tilde{P}_{j}$ as in (4.5),
$\mathbf{\tilde{X}}=\\{\tilde{X}_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,N\\}$ as in (4.7),
$\kappa_{\xi}(h)=E[X_{(n)}-X_{(1)}|\mathbf{\tilde{X}}=h]$, and
$Y_{\mu}=E[\int\nolimits_{-\infty}^{X_{1,j}}{G}_{\xi}{(u)}du/G_{\xi}(X_{1,j})]/(X_{(n)}-X_{(1)})$.
By repeating arguments in Section 3.2, (4.9) belongs to the class of
asymptotically-valid CIs and minimizes the asymptotic expected length within
this class. See Theorem A.1 in the appendix for a statement of this result.

Unfortunately, implementing the CI in (4.9) is considerably more challenging
than in the case of second-price auctions. The main reason is that there is no
closed-form expression for the PDF of $\mathbf{\tilde{X}}$ available for
first-price auctions. Without these, we cannot easily compute
$\kappa_{\xi}(h)f_{\tilde{\mathbf{X}}|\xi}(h)$ and
$f_{(Y_{\mu},\tilde{\mathbf{X}})|\xi}(y,h)$ for
$(y,h)\in\mathbb{R}\times\Sigma$. To sidestep this issue, we use a numerical
approximation to the problem based on a Taylor series expansion. See Section
A.3 for a detailed explanation.

###  4.2 Inference about the seller’s expected revenue

We now conduct inference on the average of the seller’s expected revenue using
transaction prices. Since bidders act according to (4.1) and the transaction
price equals the highest bid, the average of the seller’s expected revenue is
given by

$\pi_{K}~{}=~{}\frac{1}{n}\sum_{j=1}^{n}E\Big{[}V_{(1),j}-\tfrac{\int_{-\infty}^{V_{(1),j}}{F_{V}(u)^{K_{j}-1}}du}{F_{V}(V_{(1),j})^{K_{j}-1}}\Big{]}.$
(4.10)

Reiterating arguments in Section 3.2, we propose a CI for $\mu_{K}$ given by

$\displaystyle U(\mathbf{P})~{}=~{}$ $\displaystyle P_{(1)}+(P_{(n)}-P_{(1)})$
$\displaystyle\times$
$\displaystyle\bigg{\\{}y\in\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}\int_{\xi\in\Xi}\kappa_{\xi}(\mathbf{\tilde{P}})f_{\tilde{\mathbf{X}}|\xi}(\mathbf{\tilde{P}})dW(\xi)\leq\int_{\xi\in\Xi}f_{(Y_{\pi},\tilde{\mathbf{X}})|\xi}(y,\mathbf{\tilde{P}})d\Lambda(\xi)\bigg{\\}},$
(4.11)

where
$Y_{\pi}=E[X_{1,j}-{\int_{-\infty}^{X_{1,j}}G_{\xi}(h)dh}/{G_{\xi}(X_{1,j})}]/(X_{(n)}-X_{(1)})$
and the rest of the objects are as in Section 4.1. By the same arguments in
Section 3.2, (4.9) belongs to the class of asymptotically valid CIs and
minimizes the asymptotic expected length within this class. See Theorem A.2 in
the appendix for a statement of this result.

Implementing the CI in (4.11) suffers from the computational issues elaborated
in Section 4.1, which we avoid via numerical approximations. Once again, see
Section A.3 for details.

###  4.3 Inference about the tail index

Finally, we consider the problem of inference about the tail index using
transaction prices from first-price auctions. As in Section 3.3, we are
interested in the following hypothesis test:

$H_{0}\mathrel{\mathop{\mathchar
58\relax}}\xi\in\xi_{0}~{}~{}~{}\text{v.s.}~{}~{}~{}H_{1}\mathrel{\mathop{\mathchar
58\relax}}\xi\in\Xi_{1},$ (4.12)

where $\xi_{0}$ is a fixed parameter value in $\Xi$ and
$\Xi_{1}=\Xi\backslash\\{\xi_{0}\\}$. For brevity, we only focus on the case
when the alternative hypothesis in (4.12) is composite. By our previous
arguments, we propose using the feasible version of the generalized likelihood
ratio test in the limiting problem, i.e.,

$\varphi_{K}^{\ast}(\mathbf{P})~{}\equiv~{}1\bigg{[}{\int_{\Xi_{1}}f_{\mathbf{\tilde{X}}|\xi}(\mathbf{\tilde{P}})dW(\xi)}/{f_{\mathbf{\tilde{X}}|\xi_{0}}(\mathbf{\tilde{P}})}>q(\xi_{0},W,\alpha)\bigg{]},$
(4.13)

where $\mathbf{\tilde{P}}$, $\mathbf{\tilde{X}}$, and $\kappa_{\xi}(h)$ are as
in Section 4.1, $W$ is the user-defined weight function on $\Xi_{1}$, and
$q(\xi_{0},W,\alpha)$ is the critical value of the likelihood ratio test in
(4.12), i.e.,

$q(\xi_{0},W,\alpha)~{}\equiv~{}(1-\alpha)\text{-quantile of
}{\int_{\Xi_{1}}f_{\mathbf{\tilde{X}}|\xi_{0}}(\mathbf{\tilde{X}}_{0})dW(\xi)}/{f_{\mathbf{\tilde{X}}|\xi_{0}}(\mathbf{\tilde{X}}_{0})}.$

and $\mathbf{\tilde{X}}_{0}$ is a random vector with PDF
$f_{\mathbf{\tilde{X}}|\xi_{0}}$. By our previous arguments, it follows that
(4.13) is asymptotically valid and asymptotically level $\alpha$, and
efficient in the sense of maximizing the asymptotic weighted average power
criterion in the class of asymptotically valid and equivariant tests. See
Theorem A.3 in the appendix for a statement of this result. The implementation
of the hypothesis test in (4.13) suffers from the aforementioned computational
issues, and is addressed by the approximation described in Section A.3.

To conclude, we note that the arguments presented in Section 3.3.3 indicate
that the hypothesis test in (4.12) can be applied to test the standard
regularity conditions in the auctions literature. This can be achieved by
conducting the hypothesis test in (4.12) with $\xi_{0}=-1$ and
$\Xi_{1}=(-1,0.5]$.

##  5 Monte Carlo simulations

This section investigates the finite sample performance of our proposed
inference methods using Monte Carlo simulations. We also compare their
performance with other inference methods that rely on the standard asymptotic
framework with a divergent number of auctions. For brevity, we concentrate on
simulations for second-price auctions. The corresponding results for first-
price auctions are provided in the appendix.

We first consider the problem of inference about the winner’s expected utility
using only transaction prices. Each simulated dataset consists of $n$
independent second-price auctions where auction $j=1,\dots,n$ has $K$ bidders
with independent valuations distributed according to a distribution $F_{V}$.
As explained in Section 3.1, our parameter of interest is the average winner’s
expected utility across auctions, given by
$\mu_{K}=\frac{1}{n}\sum_{j=1}^{n}E[V_{(1),j}-V_{(2),j}]$. We conduct
simulations with four distributions for $F_{V}$: (a) the standard uniform
distribution over $[0,3]$, (b) the absolute value of standard Normal, (c) the
absolute value of Student-t(20), and (d) the Pareto distribution with exponent
0.25. These distributions satisfy condition (2.1) with $\xi=-1$, $0$, $0.05$,
and $0.25$, respectively.

Our simulations consider $n\in\\{10,100\\}$ and $K\in\\{10,100\\}$. Thus, our
simulation design includes combinations of $n$ and $K$ which are better
approximated by our asymptotic framework with a growing number of bidders
(e.g., $n=10$ and $K=100$), but also others that are better aligned to the
traditional asymptotic framework with an increasing number of auctions (e.g.,
$n=100$ and $K=10$). Our results are based on $S=500$ independent datasets.
Finally, we set the significance level equal to $\alpha=5\%$ throughout this
section.

We now describe the three CIs that we consider for $\mu_{K}$:

  1. (i)

This is our proposed CI in Section 3.1. The method is implemented by
constructing $U(\mathbf{P})$ in (3.24) with $\Xi=[-1,0.5]$ and $W$ equal to
the uniform distribution over this interval. As already explained, the
validity of this CI is based on asymptotics as $K\to\infty$.

  2. (ii)

A CI based on observing the two highest bidders in each auction. We note that
this approach is infeasible in our data setting (we only observe transaction
prices), but we take it as a benchmark for any method that relies on the
traditional asymptotics with $n\to\infty$. These data allow us to compute
$D_{j}\equiv B_{(1),j}-B_{(2),j}$ for each auction $j=1,\dots,n$. In turn,
this enables us to test $H_{0}\mathrel{\mathop{\mathchar 58\relax}}\mu_{K}=b$
using the t-statistic $\sqrt{n}(\bar{D}_{n}-b)/s_{n}$ with
$\bar{D}_{n}=\sum_{j=1}^{n}D_{j}/n$ and
$s_{n}^{2}={\sum^{n}_{j=1}(D_{j}-\bar{D}_{n})^{2}/(n-1)}$. One can then
construct a CI by standard asymptotic approximations based on $n\to\infty$.

  3. (iii)

A bootstrap-based CI. To implement this idea, we first need to express
$\mu_{K}$ as a function of the distribution of transaction prices (further
details are provided in Section A.5). By replacing this distribution with a
suitable estimator, one can consistently estimate $\mu_{K}$. In addition, one
can repeat this process using bootstrap samples to construct a CI for
$\mu_{K}$. The method is implemented with 500 bootstrap samples. The validity
of this method relies on standard asymptotic arguments with $n\to\infty$. For
references using this type of CI, see Bajari and Hortacsu (2003) and Haile and
Tamer (2003).

Table 1 presents the average coverage and length of the three CIs. We
summarize the findings as follows. First, our proposed CI exhibits excellent
coverage properties across all data configurations. As one would expect, in
scenarios where $K$ is significantly larger than $n$, our empirical CI
coverage closely approximates the desired level. On the other hand, when $n$
is considerably larger than $K$, our proposed CI exhibits slight overcoverage,
especially when valuations are distributed according to $U(0,3)$. Second, the
CI based on the two highest bids and $n\to\infty$ are relatively shorter and
tend to suffer from slight undercoverage, especially when $n$ is small or
valuations are Pareto distributed. Finally, the bootstrap-based CI tends to
suffer significant undercoverage problems. We attribute this to the fact that
$n\in\\{10,100\\}$ is insufficient to guarantee accuracy based on asymptotic
arguments as $n\to\infty$.

# Bidders | $K=10$ | $K=100$
---|---|---
# Auctions | $n=10$ | $n=100$ | $n=10$ | $n=100$
Dist. | Cov | Lgth | Cov | Lgth | Cov | Lgth | Cov | Lgth
Method (i): Our proposed CI, asy. with $K\to\infty$
$U(0,3)$ | 0.98 | 0.72 | 0.96 | 0.09 | 0.98 | 0.30 | 0.99 | 0.13
$|N(0,1)|$ | 0.95 | 1.54 | 0.97 | 0.29 | 0.94 | 1.20 | 0.98 | 0.23
$|t(20)|$ | 0.93 | 1.74 | 0.97 | 0.37 | 0.94 | 1.52 | 0.98 | 0.35
$Pa(0.25)$ | 0.96 | 1.45 | 0.99 | 0.62 | 0.95 | 2.62 | 0.98 | 1.11
Method (ii): CI based on two highest bids, asy. with $n\to\infty$
$U(0,3)$ | 0.86 | 0.28 | 0.94 | 0.10 | 0.83 | 0.03 | 0.93 | 0.01
$|N(0,1)|$ | 0.89 | 0.46 | 0.94 | 0.16 | 0.88 | 0.35 | 0.96 | 0.12
$|t(20)|$ | 0.89 | 0.55 | 0.92 | 0.19 | 0.86 | 0.48 | 0.92 | 0.17
$Pa(0.25)$ | 0.81 | 0.76 | 0.90 | 0.30 | 0.80 | 1.34 | 0.90 | 0.51
Method (iii): CI based on bootstrap, asy. with $n\to\infty$
$U(0,3)$ | 0.66 | 0.22 | 0.93 | 0.08 | 0.67 | 0.03 | 0.95 | 0.01
$|N(0,1)|$ | 0.04 | 0.22 | 0.23 | 0.11 | 0.05 | 0.15 | 0.20 | 0.08
$|t(20)|$ | 0.02 | 0.24 | 0.15 | 0.13 | 0.04 | 0.19 | 0.13 | 0.11
$Pa(0.25)$ | 0.00 | 0.17 | 0.06 | 0.15 | 0.01 | 0.31 | 0.05 | 0.25
Table 1: Empirical coverage frequency (Cov) and length (Lgth) of various CIs
for the winner’s expected utility $\mu_{K}$ in second-price auctions. The
results are the average of 500 simulation draws and a nominal coverage level
of 95%.

Next, we consider the problem of hypothesis testing the value of the tail
index based on transaction prices. Motivated by Section 3.3.3, we focus on
$H_{0}\mathrel{\mathop{\mathchar 58\relax}}\xi=-1$ vs.
$H_{1}\mathrel{\mathop{\mathchar 58\relax}}\xi\in(-1,0.5]$, with $W$ equal to
the uniform distribution over this interval. As explained, the validity of
this method is based on asymptotics as $K\to\infty$. We consider the same four
distributions for $F_{V}$: (a) the standard uniform distribution over $[0,3]$,
(b) the absolute value of standard Normal, (c) the absolute value of
Student-t(20), and (d) the Pareto distribution with exponent 0.25. The first
distribution satisfies $H_{0}\mathrel{\mathop{\mathchar 58\relax}}\xi=-1$,
while the other three belong to $H_{1}$ with $\xi=0$, $0.05$, and $0.25$,
respectively.

Table 2 presents the empirical rejection rate of the test proposed in Section
3.3.3 over 500 simulations using a nominal size of $5\%$. Under $H_{0}$ (i.e.,
when valuations are $U(0,3)$), our proposed test controls size as long as the
number of bidders is not too small relative to the sample size. Under $H_{1}$
(i.e., when valuations are not $U(0,3)$), our methodology provides reasonable
power.

# Bidders | 10 | 20 | 100
---|---|---|---
# Auctions | 10 | 100 | 10 | 100 | 10 | 100
$U(0,3)$ | 0.06 | 0.35 | 0.05 | 0.14 | 0.04 | 0.05
$|N(0,1)|$ | 0.42 | 1.00 | 0.46 | 1.00 | 0.48 | 1.00
$|t(20)|$ | 0.46 | 1.00 | 0.48 | 1.00 | 0.54 | 1.00
$Pa(0.25)$ | 0.73 | 1.00 | 0.71 | 1.00 | 0.68 | 1.00
Table 2: Empirical rejection rate of the test proposed in Section 3.3.3 for
$H_{0}\mathrel{\mathop{\mathchar 58\relax}}\xi=-1$ in second-price auctions.
The results are the average of 500 simulation draws and a nominal size of 5%.

##  6 Empirical application

This section illustrates our new inference method using auction data for
vehicle license plates in Hong Kong. Since 1973, the Hong Kong government has
employed standard oral ascending auctions to auction license plates. As
mentioned in Section 3, this auction format is weakly equivalent to a second-
price auction. The dataset was provided by Ng et al. (2010), who obtained it
from the Hong Kong Transport Department. The data include a detailed
description of 292 license plate auctions conducted from 1997 to 2009,
resulting in the sale of a total of 40,000 license plates. Given the
substantial size of the auction venue and the rapid increase in the number of
registered private cars in Hong Kong, it is reasonable to assume that a
significant number of bidders participated in these auctions. On average, each
one of these auctions sells more than a hundred different plates sequentially.
In terms of our asymptotic framework, we treat each one of these license
plates as an individual instance of a second-price auction with a large number
of bidders.

While the dataset includes numerous individual instances of license plate
auctions, there is significant heterogeneity among them. In fact, the main
finding in Ng et al. (2010) is that certain plates are more valuable due to
superstition. To account for this observed heterogeneity, we focus on ordinary
license plates that meet specific criteria: letters are not ‘HK’, letters are
not the same (e.g., not AA, BB, CC, etc.), the numbers on the plate are not in
order (e.g., not 1369), or in reverse order (e.g., not 9631), the number part
of the plate has 4 digits, and none of these digits is an 8 or a 4 (considered
fortunate or unfortunate), and the transaction price exceeds the reserve
price. After imposing these restrictions, we are left with 318 license plates
sold from 1997 until 2008. These data are further divided by year, allowing
valuation distributions to vary over time. This results in 12 separate
datasets, one for each year from 1997 to 2008, with an average of 26.5
auctions per year. We assume that the selected auctions within each year are
homogeneous, with a large and relatively constant number of bidders. Under
these conditions, we can implement our methods for each one of these years.

Table 3 displays the confidence intervals for the expected utility of the
winner and the $p$-values of the test (3.35) with a null hypothesis
$H_{0}\mathrel{\mathop{\mathchar 58\relax}}\xi=-1$ using data from each year.
Our analysis reveals several interesting empirical observations. First, the
expected utility of the winner is economically substantial, with the midpoints
of the 95% confidence intervals ranging from approximately 1,500 to 7,000 HKDs
(equivalent to 192 to 895 USD at the current exchange rate). Prior to 2006,
the average confidence interval had a midpoint of approximately 3,500 HKDs and
a width of around 4,500 HKDs. After 2006, these figures were reduced to 1,600
HKDs and 2,300 HKDs, respectively. We speculate that these differences could
be attributed to the introduction of special license plates in these auctions
in 2006. Third, the hypothesis of $\xi=-1$ is strongly rejected for all years.
This suggests that the standard regularity conditions imposed by traditional
inference methods in auction literature do not hold in this dataset.

year | $n$ | 95% CI |  $p$-value for |  | year | $n$ | 95% CI |  $p$-value for
---|---|---|---|---|---|---|---|---
for $\mu_{K}$ | $H_{0}\mathrel{\mathop{\mathchar 58\relax}}\xi=-1$ |  | for $\mu_{K}$ | $H_{0}\mathrel{\mathop{\mathchar 58\relax}}\xi=-1$
1997 | 26 | [2.23 , 9.23] | 0.00 |  | 2003 | 46 | [0.82 , 2.98] | 0.00
1998 | 27 | [1.71 , 7.29] | 0.00 |  | 2004 | 12 | [0.69 , 5.59] | 0.00
1999 | 32 | [1.22 , 5.72] | 0.00 |  | 2005 | 22 | [0.61 , 3.83] | 0.00
2000 | 29 | [1.58 , 6.63] | 0.00 |  | 2006 | 17 | [0.12 , 1.69] | 0.00
2001 | 34 | [1.23 , 4.71] | 0.00 |  | 2007 | 31 | [0.69 , 3.07] | 0.00
2002 | 18 | [1.02 , 6.30] | 0.00 |  | 2008 | 24 | [0.58 , 3.54] | 0.00
Table 3: 95% CIs for the winner’s expected utility (in 1,000 HKDs) and
$p$-values for the test in (3.35) for $H_{0}\mathrel{\mathop{\mathchar
58\relax}}\xi=-1$ in Hong Kong car license plate auctions.

##  7 Conclusions

This paper considers the problem inference for first-price or second-price
sealed-bid auctions with many bidders with symmetric independent private
values. In this context, we present a novel asymptotic framework where the
number of bidders increases significantly while the number of auctions remains
small and fixed. This approach differs from the more conventional approach
with a divergent number of auctions, while the number of bidders remains
fixed. We argue that our results provide an accurate approximation in auction
settings where the number of bidders is large relative to the number of
auctions. This framework is especially well-suited for applications with
substantial heterogeneity across auctions, where only a few truly homogeneous
auctions exist.

Within our novel asymptotic framework, we introduce new inference methods for
the expected utility of the auction winner, the expected revenue for the
seller, and the tail behavior of the valuation distribution. We show that the
latter can serve as a means to test the validity of the regularity conditions
typically imposed in auction literature (e.g., bounded valuation support and
nonzero density at the upper end of the support). Our data requirements are
minimal; our tests only necessitate observing transaction prices from a fixed
and finite number of auctions. That is, we do not require observing multiple
bids from these auctions or the number of bidders.

Our methodology relies on the fact that, as the number of bidders grows, the
transaction prices reveal the tail properties of the valuation distribution.
This information is sufficient to provide asymptotically valid inference for
the above-mentioned objects of economic interest. Within our asymptotic
framework, our inference methods are shown to control size and have desirable
power properties. We demonstrate through Monte Carlo simulations that our
methodology provides an accurate approximation in finite samples.

##  Appendix A Appendix

The appendix is organized as follows. Section A.1 collects all of the proofs
omitted from the main text. Section A.2 gives intermediate results. Section
A.3 provides computational details omitted from the main text. Section A.4
presents additional Monte Carlo simulations, focusing on first-price auctions.
Finally, Section A.5 provides auxiliary derivations related to the Monte Carlo
simulations.

###  A.1 Proofs of results in the main text

###### Proof of Lemma 2.1.

By independence across auctions, it suffices to prove that for any auction
$j=1,\ldots n$ and as $K\to\infty$,

$\displaystyle\big{(}\tfrac{V_{(1),j}-b_{K}}{a_{K}},\dots,~{}\tfrac{V_{(d),j}-b_{K}}{a_{K}}\big{)}~{}\overset{d}{\to}~{}\big{(}H_{\xi}(E_{1,j}),\dots,~{}H_{\xi}\big{(}\sum\nolimits_{s=1}^{d}E_{s,j}\big{)}\big{)},$
(A.1)

where $\\{E_{s,j}\mathrel{\mathop{\mathchar 58\relax}}s=1,\dots,d\\}$ are
i.i.d. standard exponential random variables.

Let $x$ be any continuity point of $G_{\xi}$. Since $K_{j}\to\infty$ as
$K\to\infty$, (2.1) implies that there is a sequence of constants
$\\{(a_{K_{j}},b_{K_{j}})\in\mathbb{R}_{++}\times\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}K_{j}\in\mathbb{N}\\}$ such that

$\lim_{K\to\infty}F(a_{K_{j}}x+b_{K_{j}})^{K_{j}}~{}=~{}G_{\xi}(x)~{}\text{ as
}K\to\infty.$ (A.2)

Under (A.2), de Haan and Ferreira (2006, Theorem 2.1.1), implies that as
$K\to\infty$,

$\displaystyle\Psi_{j}\equiv\big{(}\tfrac{V_{(1),j}-b_{K_{j}}}{a_{K_{j}}},\dots,~{}\tfrac{V_{(d),j}-b_{K_{j}}}{a_{K_{j}}}\big{)}~{}\overset{d}{\to}~{}\big{(}H_{\xi}(E_{1,j}),\dots,~{}H_{\xi}\big{(}\sum\nolimits_{s=1}^{d}E_{s,j}\big{)}\big{)}.$
(A.3)

Note that

$\displaystyle\big{(}\tfrac{V_{(1),j}-b_{K}}{a_{K}},\dots,\tfrac{V_{(d),j}-b_{K}}{a_{K}}\big{)}$
$\displaystyle~{}=~{}\Psi_{j}\tfrac{a_{K_{j}}}{a_{K}}~{}+~{}\big{(}\tfrac{b_{K}-b_{K_{j}}}{a_{K}},\dots,\tfrac{b_{K}-b_{K_{j}}}{a_{K}}\big{)}.$
(A.4)

By (A.3) and (A.4), (A.1) follows from showing that

$\lim_{K\to\infty}\big{(}\tfrac{a_{K_{j}}}{a_{K}},~{}\tfrac{b_{K_{j}}-b_{K}}{a_{K}}\big{)}~{}=~{}(1,~{}0).$
(A.5)

We devote the remainder of this proof to establish (A.5).

Let $x$ be any continuity point of $G_{\xi}$. By (A.2) and $K_{j}/K\to 1$ as
$K\to\infty$,

$\lim_{K\to\infty}F(a_{K_{j}}x+b_{K_{j}})^{K}~{}=~{}G_{\xi}(x)~{}\text{ as
}K\to\infty.$ (A.6)

Under (2.1) and (A.6), de Haan (1976, Lemma 1) implies that for all
$y\in\mathbb{R}$,

$G_{\xi}\big{(}\lim_{K\to\infty}\big{(}\tfrac{a_{K_{j}}}{a_{K}}y+\tfrac{b_{K_{j}}-b_{K}}{a_{K}}\big{)}\big{)}~{}=~{}G_{\xi}(y).$
(A.7)

Regardless of $\xi$, there is a continuum of values for which $G_{\xi}$ is
strictly increasing and, thus, invertible. For any $y$ in this continuum,
(A.7) implies that

$\lim_{K\to\infty}\tfrac{a_{K_{j}}}{a_{K}}y+\tfrac{b_{K_{j}}-b_{K}}{a_{K}}~{}=~{}y.$
(A.8)

From this observation, (A.5) follows.

###### Proof of Lemma 3.1.

This result follows from Lemma 2.1, Lemma A.1, and the continuous mapping
theorem.

###### Proof of Theorem 3.1.

Part 1. By (3.2) and Lemma 2.1, as $K\to\infty$,

$\big{\\{}\big{(}\tfrac{V_{(1),j}-P_{j}}{a_{K}},\tfrac{P_{j}-b_{K}}{a_{K}}\big{)}\mathrel{\mathop{\mathchar
58\relax}}j=1,\ldots,N\big{\\}}~{}\overset{d}{\to}~{}\\{(Z_{1,j}-Z_{2,j},Z_{2,j})\mathrel{\mathop{\mathchar
58\relax}}j=1,\ldots,N\\}.$ (A.9)

Let $\delta>0$ and $\bar{K}\in\mathbb{N}$ be as in Lemma A.2. Then, for all
$j=1,\dots,N$ and $K>\bar{K}$,

$E\big{[}|\tfrac{V_{(1),j}-P_{j}}{a_{K}}|^{1+\delta}\big{]}~{}\overset{(1)}{=}~{}E\big{[}|\tfrac{V_{(1),j}-V_{(2),j}}{a_{K}}|^{1+\delta}\big{]}~{}\leq~{}2^{\delta}\big{(}E\big{[}|\tfrac{V_{(1),j}-b_{K}}{a_{K}}|^{1+\delta}+|\tfrac{V_{(2),j}-b_{K}}{a_{K}}|^{1+\delta}\big{]}\big{)}~{}\overset{(2)}{<}~{}\infty,$
(A.10)

where (1) holds by (3.1) and (3.2), and (2) by Lemma A.2. By (3.16), (A.9),
and (A.10), as $K\to\infty$,

$\big{\\{}\tfrac{\mu_{K}}{a_{K}},\big{\\{}\big{(}\tfrac{P_{j}-b_{K}}{a_{K}}\big{)}\mathrel{\mathop{\mathchar
58\relax}}j=1,\ldots,N\big{\\}}\big{\\}}~{}\overset{d}{\to}~{}\\{E[Z_{1,1}-Z_{2,1}],\\{Z_{2,j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\ldots,N\\}\\},$ (A.11)

where we have used that $Z_{1,1}-Z_{2,1}\overset{d}{=}Z_{1,j}-Z_{2,j}$ for all
$j=1,\ldots,n$. Define

$R_{K}~{}\equiv~{}\bigg{\\{}\begin{array}[]{cc}\tfrac{\mu_{K}}{P_{(n)}-P_{(1)}}&\text{if
}P_{(n)}\not=P_{(1)},\\\ 0&\text{if }P_{(n)}=P_{(1)}.\end{array}$ (A.12)

By (A.11), (A.12), Lemma A.1, and the continuous mapping theorem, we have
that, as $K\to\infty$,

$\\{R_{K},\mathbf{\tilde{P}}\\}~{}\overset{d}{\to}~{}\\{Y_{\mu},\mathbf{\tilde{Z}}\\}.$
(A.13)

Next, consider the following derivation as $K\to\infty$.

$\displaystyle P(\mu_{K}\in U(\mathbf{P}))$
$\displaystyle~{}=~{}P(\\{\mu_{K}\in
U(\mathbf{P})\\}\cap\\{P_{(n)}\not=P_{(1)}\\})+P(\\{\mu_{K}\in
U(\mathbf{P})\\}\cap\\{P_{(n)}=P_{(1)}\\})$
$\displaystyle~{}\overset{(1)}{=}~{}P(\\{\mu_{K}\in(P_{(n)}-P_{(1)})\times\tilde{U}(\mathbf{\tilde{P}})\\}\cap\\{P_{(n)}\not=P_{(1)}\\})+o(1)$
$\displaystyle~{}\overset{(2)}{=}~{}P(\\{R_{K}\in\tilde{U}(\mathbf{\tilde{P}})\\}\cap\\{P_{(n)}\not=P_{(1)}\\})+o(1)$
$\displaystyle~{}\overset{(3)}{=}~{}P(R_{K}\in\tilde{U}(\mathbf{\tilde{P}}))+o(1)$
$\displaystyle~{}=~{}P(\\{R_{K},\mathbf{\tilde{P}}\\}\in\\{(y,h)\in\mathbb{R}\times\Sigma\mathrel{\mathop{\mathchar
58\relax}}y\in\tilde{U}(h)\\})+o(1)$
$\displaystyle~{}\overset{(4)}{\to}~{}P(\\{Y_{\mu},\mathbf{\tilde{Z}}\\}\in\\{(y,h)\in\mathbb{R}\times\Sigma\mathrel{\mathop{\mathchar
58\relax}}y\in\tilde{U}(h)\\})$
$\displaystyle~{}=~{}P(Y_{\mu}\in\tilde{U}(\mathbf{\tilde{Z}})),$

as desired, where (1) holds by (3.17) and Lemma A.1, (2) holds because (A.12)
implies that
$\\{\mu_{K}\in(P_{(n)}-P_{(1)})\times\tilde{U}(\mathbf{\tilde{P}})\\}=\\{R_{K}\in\tilde{U}(\mathbf{\tilde{P}})\\}$
under $P_{(n)}\not=P_{(1)}$, (3) by Lemma A.1, and (4) by (A.13), Assumption
(a), and the Portmanteau Theorem.

Part 2. By similar arguments as those used in the proof of Lemma 3.1, we have
that, as $K\to\infty$,

$\\{{(P_{(n)}-P_{(1)})}/{a_{K}},\mathbf{\tilde{P}}\\}~{}\overset{d}{\to}~{}\\{Z_{(n)}-Z_{(1)},\mathbf{\tilde{Z}}\\}.$
(A.14)

By (A.14), Assumption (c), and the continuous mapping theorem, we have that,
as $K\to\infty$,

${((P_{(n)}-P_{(1)})}/{a_{K}})\times\lg(\tilde{U}(\mathbf{\tilde{P}}))~{}\overset{d}{\to}~{}(Z_{(n)}-Z_{(1)})\times\lg(\tilde{U}(\mathbf{\tilde{Z}})).$
(A.15)

Let $\delta>0$ and $\bar{K}\in\mathbb{N}$ be as in Lemma A.2. Then,

$\displaystyle|P_{(n)}-P_{(1)}|^{1+\delta}~{}$
$\displaystyle\leq~{}(\sum_{l,m=1}^{n}|P_{l}-P_{m}|)^{1+\delta}$
$\displaystyle\overset{(1)}{=}~{}(\sum_{l,m=1}^{n}|V_{(2),l}-V_{(2),m}|)^{1+\delta}$
$\displaystyle{\leq}~{}(2n)^{1+\delta}\sum_{l=1}^{n}|V_{(2),l}-b_{K}|^{1+\delta},$
(A.16)

where (1) holds by (3.2). Then, for all $K>\bar{K}$,

$\displaystyle
E[|\tfrac{(P_{(n)}-P_{(1)})}{a_{K}}\lg(\tilde{U}(\mathbf{\tilde{P}}))|^{1+\delta}]~{}\overset{(1)}{\leq}~{}(2n)^{1+\delta}\sum_{l=1}^{n}E\big{[}|\tfrac{V_{(2),l}-b_{K}}{a_{K}}|^{1+\delta}|\lg(\tilde{U}(\mathbf{\tilde{P}}))|^{1+\delta}\big{]}~{}\overset{(2)}{<}~{}\infty,$
(A.17)

where (1) holds by (A.16), and (2) by Assumption (b), $\lg(\tilde{U}(h))\geq
0$ for all $h\in\Sigma$, and Lemma A.2.

Then, consider the following derivation as $K\to\infty$.

$\displaystyle E[\lg(U(\mathbf{P}))/a_{K}]~{}$
$\displaystyle\overset{(1)}{=}~{}E[((P_{(n)}-P_{(1)})/a_{K})\times\lg(\tilde{U}(\mathbf{\tilde{P}}))]$
$\displaystyle\overset{(2)}{\to}~{}E[(Z_{(n)}-Z_{(1)})\times\lg(\tilde{U}(\mathbf{\tilde{Z}}))]$
$\displaystyle=~{}E_{\xi}[\kappa_{\xi}(\mathbf{\tilde{Z}})\times\lg(\tilde{U}(\mathbf{\tilde{Z}}))],$

as desired, where (1) holds by (3.17), and (2) by (A.15) and (A.17).

###### Proof of Theorem 3.2.

This proof follows from the same arguments as in Theorem 3.1. The main
difference between the proofs is that the vector
$\\{(V_{(1),j}-P_{j})\mathrel{\mathop{\mathchar 58\relax}}j=1,\ldots,N\\}$ is
replaced by $\\{(P_{j}-b_{K})\mathrel{\mathop{\mathchar
58\relax}}j=1,\ldots,N\\}$, which then implies that $(\pi_{K}-b_{K})$ and
$Y_{\mu}$ are replaced by $\mu_{K}$ and $Y_{\pi}$, respectively. With these
changes in place, the desired result follows from repeating the steps used to
prove Theorem 3.1.

###### Proof of Theorem 3.3.

This proof follows from applying Müller (2011, Theorem 1) based on the
convergence result in Lemma 3.1. To apply these results in that paper, we
specify the connection between our object and the relevant objects in that
paper: the class of DGPs satisfying (2.1) take the role of $\mathcal{M}$,
$K\to\infty$ takes the role of $T\to\infty$, $j=1,\ldots,n$ the role of
$i=1,\ldots,n$, $\\{\xi_{0}\\}$ the role of $\Theta_{0}$, $\\{\xi_{1}\\}$ the
role of $\Theta_{1}$, $\mathbf{{P}}$ the role of $Y_{T}$, $\mathbf{\tilde{P}}$
the role of $X_{T}$, $\Sigma$ the role of $S$, the self-normalizing
transformation
$\\{(P_{j}-P_{(1)})/(P_{(n)}-P_{(1)})\\}_{j=1}^{N}\mathrel{\mathop{\mathchar
58\relax}}\mathbf{P}\to\Sigma$ the role of $h_{T}$, $\mathbf{\tilde{Z}}$ the
role of $X$,
$\tilde{\varphi}^{\ast}(\mathbf{\tilde{Z}})\equiv{\varphi}^{\ast}(\mathbf{{Z}})$
the role of $\varphi_{S}(Y)$, $\varphi_{K}^{\ast}(\mathbf{P})$ the role of
$\hat{\varphi}_{T}^{\ast}(\mathbf{P})$, $\varphi_{K}(\mathbf{P})$ the role of
$\varphi_{T}(\mathbf{P})$, and $f_{\mathbf{\tilde{Z}}|\xi_{0}}$ the role of
$\mu_{P}$. Moreover, we set $\mathcal{F}_{s}$ equal to the set with the zero
function and $\pi_{0}=0$, which effectively trivializes Müller (2011,
Equations (5)-(6) and (10)-(11)).

By definition,
$\tilde{\varphi}^{\ast}(\mathbf{\tilde{Z}})\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to[0,1]$ is a level-$\alpha$ test in the limiting problem,
i.e., (3.34) holds. According to the Neyman-Pearson Lemma, it maximizes power
in the limiting problem. Finally, note that $\tilde{\varphi}^{\ast}$ is
continuous except for a set of zero $f_{\mathbf{\tilde{Z}}|\xi_{0}}$-measure.
These conditions align with the requirements in Müller (2011, Theorem 1), and
so our desired results follow immediately from its application.

###### Proof of Theorem 3.4.

This proof follows from the same arguments as in Theorem 3.3.

###### Proof of Lemma 3.2.

Fix $x>0$ arbitrarily. By $v_{H}<\infty$ and de Haan and Ferreira (2006,
Theorem 1.2.1), it suffices show that

$\lim_{t\downarrow 0}\tfrac{1-F_{V}(v_{H}-tx)}{1-F_{V}(v_{H}-t)}=x.$

To this end, consider the following derivation.

$\lim_{t\downarrow
0}\tfrac{1-F_{V}(v_{H}-tx)}{1-F_{V}(v_{H}-t)}~{}\overset{(1)}{=}~{}\lim_{t\downarrow
0}\tfrac{xf_{V}(v_{H}-tx)}{f_{V}(v_{H}-t)}~{}\overset{(2)}{=}~{}x,$

as desired, where (1) holds by L’Hôpital rule and (2) by $f_{V}(v)\to
f_{V}(v_{H})>0$ as $v\uparrow v_{H}$.

###### Proof of Lemma 4.1.

This result follows from Lemma A.1, Lemma A.7, and the continuous mapping
theorem.

###  A.2 Auxiliary results

######  Lemma A.1.

For any $n>1$ and for first-price or second-price auctions,

$P(P_{(n)}=P_{(1)})~{}=~{}0.$

###### Proof.

Note that

$\\{P_{(n)}=P_{(1)}\\}~{}\overset{(1)}{\subseteq}~{}\underset{l,m=1,\ldots,n,l\neq
m}{\bigcap}~{}\underset{i_{l}=1,\ldots,K_{l},i_{m}=1,\ldots,K_{m}}{\bigcup}\\{V_{i_{l},l}=V_{i_{m},m}\\},$
(A.18)

where (1) holds by (3.2) and (4.1) for second-price and first-price auctions,
respectively. Then,

$P(P_{(n)}=P_{(1)})~{}\overset{(1)}{\leq}~{}\min_{l,m=1,\ldots,n,l\neq
m}~{}\sum_{i_{l}=1,\ldots,K_{l},i_{m}=1,\ldots,K_{m}}P(V_{i_{l},l}=V_{i_{m},m})~{}\overset{(2)}{=}~{}0,$

as desired, where (1) holds by (A.18) and (2) by the fact that valuations are
continuously distributed and $l,m$ refer to different and, thus, independent,
auctions.

######  Lemma A.2.

Assume (2.1) holds. For some $\varepsilon>0$ with $(1+\varepsilon)\xi<1$,
assume that $E[|V_{i,j}|^{1+\varepsilon}]<\infty$ for all $i=1\dots,K_{j}$ in
auction $j=1,\dots,N$. Then, $\exists\bar{K}\in\mathbb{N}$ such that for
$\delta=\varepsilon/2$ and $K\geq\bar{K}$,

$\displaystyle
E[|\tfrac{V_{(1),j}-b_{K}}{a_{K}}|^{1+\delta}]~{}<~{}\infty~{}~{}~{}~{}\text{and}~{}~{}~{}~{}E[|\tfrac{V_{(2),j}-b_{K}}{a_{K}}|^{1+\delta}]~{}<~{}\infty.$
(A.19)

###### Proof.

The first result in (A.19) is a corollary of de Haan and Ferreira (2006,
Theorem 5.3.1). Thus, we focus the rest of this proof on the second result in
(A.19).

It is convenient to introduce the following notation. For any
$\tilde{K}\in\mathbb{N}$, let $U({\tilde{K}})$ distributed according to
$\max\\{V_{i,j}\mathrel{\mathop{\mathchar 58\relax}}j=1,\dots,\tilde{K}\\}$.
For any $x\in[v_{L},v_{H}]$, the CDF and PDF of $U({\tilde{K}})$ are

$F_{U{(\tilde{K})}}(x)~{}=~{}F_{V}(x)^{\tilde{K}}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}f_{U({\tilde{K}})}(x)~{}=~{}\tilde{K}(F_{V}(x))^{\tilde{K}-1}f_{V}(x).$
(A.20)

Conditional on $V_{(1),j}=x\in[v_{L},v_{H}]$, $V_{(2),j}$ is the highest
valuation among the remaining $K_{j}-1$ bidders. Thus, for any $t\leq x$ with
$x,t\in[v_{L},v_{H}]$, $P(V_{(2),j}\leq
t|V_{(1),j}=x)=(F_{V}(t)/F_{V}(x))^{K_{j}-1}$. By setting
$\tilde{t}=(t-b_{K})/a_{K}$ and $\tilde{x}=(x-b_{K})/a_{K}$ for any
$x,t\in[v_{L},v_{H}]$ with $t\leq x$,

$P\Big{(}\tfrac{V_{(2),j}-b_{K}}{a_{K}}\leq\tilde{t}\Big{|}\tfrac{V_{(1),j}-b_{K}}{a_{K}}=\tilde{x}\Big{)}~{}=~{}\Big{(}\tfrac{F_{V}(\tilde{t}a_{K}+b_{K})}{F_{V}(\tilde{x}a_{K}+b_{K})}\Big{)}^{K_{j}-1}$

and, so, the conditional PDF is

$f_{\Big{(}\tfrac{V_{(2),j}-b_{K}}{a_{K}}\Big{|}\tfrac{V_{(1),j}-b_{K}}{a_{K}}=\tilde{x}\Big{)}}(\tilde{t})~{}=~{}\tfrac{(K_{j}-1)(F_{V}(\tilde{t}a_{K}+b_{K}))^{K_{j}-2}f_{V}(\tilde{t}a_{K}+b_{K})a_{K}}{F_{V}(\tilde{x}a_{K}+b_{K})^{K_{j}-1}}~{}=~{}\tfrac{f_{U(K_{j}-1)}(\tilde{t}a_{K}+b_{K})a_{K}}{F_{U(K_{j}-1)}(\tilde{x}a_{K}+b_{K})}.$
(A.21)

For any $x\in[v_{L},v_{H}]$, we have
$P(({V_{(1),j}-b_{K}})/{a_{K}}\leq\tilde{x})=(F_{V}(\tilde{x}a_{K}+b_{K}))^{K_{j}}$
and, so,

$f_{\tfrac{V_{(1),j}-b_{K}}{a_{K}}}(\tilde{x})~{}=~{}K_{j}(F_{V}(\tilde{x}a_{K}+b_{K}))^{K_{j}-1}f_{V}(\tilde{x}a_{K}+b_{K})a_{K}~{}=~{}f_{U(K_{j})}(\tilde{x}a_{K}+b_{K})a_{K}.$
(A.22)

Consider the following derivation:

$\displaystyle
E\big{[}\big{|}\tfrac{V_{(2),j}-b_{K}}{a_{K}}\big{|}^{1+\delta}\big{]}$
$\displaystyle=~{}E\big{[}E\big{[}\big{|}\tfrac{V_{(2),j}-b_{K}}{a_{K}}\big{|}^{1+\delta}\big{|}\tfrac{V_{(1),j}-b_{K}}{a_{K}}\big{]}\big{]}$
$\displaystyle\overset{(1)}{=}~{}\int_{\tilde{x}=\tfrac{v_{L}-b_{K}}{a_{K}}}^{\tilde{x}=\tfrac{v_{H}-b_{K}}{a_{K}}}\Big{(}\int_{\tilde{t}=\tfrac{v_{L}-b_{K}}{a_{K}}}^{\tilde{t}=\tilde{x}}|\tilde{t}|^{1+\delta}\tfrac{f_{U(K_{j}-1)}(\tilde{t}a_{K}+b_{K})a_{K}}{F_{U(K_{j}-1)}(\tilde{x}a_{K}+b_{K})}d\tilde{t}\Big{)}f_{U(K_{j})}(\tilde{x}a_{K}+b_{K})a_{K}d\tilde{x}$
$\displaystyle\overset{(2)}{=}~{}\int_{\tilde{x}=\tfrac{v_{L}-b_{K}}{a_{K}}}^{\tilde{x}=\tfrac{v_{H}-b_{K}}{a_{K}}}\Big{(}E\big{[}\big{|}\tfrac{U(K_{j}-1)-b_{K}}{a_{K}}\big{|}^{1+\delta}1[U(K_{j}-1)\leq\tilde{x}a_{K}+b_{K}]\big{]}\Big{)}\tfrac{f_{U(K_{j})}(\tilde{x}a_{K}+b_{K})}{F_{U(K_{j}-1)}(\tilde{x}a_{K}+b_{K})}a_{K}d\tilde{x},$
(A.23)

where (1) holds by (A.21) and (A.22) and (2) by the change of variables
$t=\tilde{t}a_{K}+b_{K}$.

For any $p,q>1$ with $1/p+1/q=1$, we have

$\displaystyle
E\big{[}\big{|}\tfrac{U(K_{j}-1)-b_{K}}{a_{K}}\big{|}^{1+\delta}1[U(K_{j}-1)\leq\tilde{x}a_{K}+b_{K}]\big{]}$
$\displaystyle\overset{(1)}{\leq}~{}E\big{[}\big{|}\tfrac{U(K_{j}-1)-b_{K_{j}-1}}{a_{K_{j}-1}}\tfrac{a_{K_{j}-1}}{a_{K}}+\tfrac{b_{K_{j}-1}-b_{K}}{a_{K}}\big{|}^{(1+\delta)p}\big{]}^{1/p}F_{U(K_{j}-1)}(\tilde{x}a_{K}+b_{K})^{1/q}$
$\displaystyle\overset{(2)}{\leq}~{}\Bigg{\\{}\begin{array}[]{c}2^{\tfrac{(1+\delta)p-1}{p}}\Big{[}E\big{[}\big{|}\tfrac{U(K_{j}-1)-b_{K_{j}-1}}{a_{K_{j}-1}}\big{|}^{(1+\delta)p}\big{]}\big{|}\tfrac{a_{K_{j}-1}}{a_{K}}\big{|}^{(1+\delta)p}+\big{|}\tfrac{b_{K_{j}-1}-b_{K}}{a_{K}}\big{|}^{(1+\delta)p}\Big{]}^{1/p}\\\
\times F_{U(K_{j}-1)}(\tilde{x}a_{K}+b_{K})^{1/q}\end{array}\Bigg{\\}},$
(A.26)

where (1) holds by Hölder’s inequality and (2) by Minkowski’s inequality.
Next, let $C>2^{{((1+\delta)p-1)}/{p}}[E(|Z_{\xi}|^{(1+\delta)p})]^{1/p}$
where $Z_{\xi}$ has CDF $G_{\xi}$, and
$p=(1+\varepsilon)/(1+\delta)=(2+2\varepsilon)/(2+\varepsilon)>1$ (since
$\delta=\varepsilon/2$). Under these conditions, we have that

$\lim_{K\to\infty}\Big{[}E\big{[}\big{|}\tfrac{U(K_{j}-1)-b_{K_{j}-1}}{a_{K_{j}-1}}\big{|}^{(1+\delta)p}\big{]}\big{|}\tfrac{a_{K_{j}-1}}{a_{K}}\big{|}^{(1+\delta)p}+\big{|}\tfrac{b_{K_{j}-1}-b_{K}}{a_{K}}\big{|}^{(1+\delta)p}\Big{]}^{1/p}~{}<~{}C.$
(A.27)

This result is a corollary of two observations. First, we note that (A.5)
implies that
$\lim_{K\to\infty}(a_{K_{j}-1}/a_{K},(b_{K_{j}-1}-b_{K})/a_{K})=(1,0)$.
Second, under $p=(1+\varepsilon)/(1+\delta)$, $(1+\varepsilon)\xi<1$, and
$E[|V_{i,j}|^{1+\varepsilon}]<\infty$, de Haan and Ferreira (2006, Theorem
5.3.1) implies that

$\lim_{K\to\infty}E(|\tfrac{U(K_{j}-1)-b_{K_{j}-1}}{a_{K_{j}-1}}|^{(1+\delta)p})~{}=~{}\lim_{K\to\infty}E(|\tfrac{U(K_{j}-1)-b_{K_{j}-1}}{a_{K_{j}-1}}|^{1+\varepsilon})~{}=~{}E[|Z_{\xi}|^{1+\varepsilon}]~{}<~{}\infty.$

Note that (A.26) and (A.27) imply that $\exists\bar{K}\in\mathbb{N}$ such that
for all $K>\bar{K}$,

$E\big{[}\big{|}\tfrac{U(K_{j}-1)-b_{K}}{a_{K}}\big{|}^{1+\delta}1[U(K_{j}-1)\leq\tilde{x}a_{K}+b_{K}]\big{]}~{}\leq~{}C\times
F_{U(K_{j}-1)}(\tilde{x}a_{K}+b_{K})^{1/q}.$ (A.28)

To complete the proof, consider the following argument for all
$K>\max\\{\bar{K},2\\}$,

$\displaystyle E[|\tfrac{V_{(2),j}-b_{K}}{a_{K}}|^{1+\delta}]$
$\displaystyle\overset{(1)}{\leq}~{}C\int_{\tfrac{v_{L}-b_{K}}{a_{K}}}^{\tfrac{v_{H}-b_{K}}{a_{K}}}F_{U(K_{j}-1)}(\tilde{x}a_{K}+b_{K})^{1/q-1}f_{U(K_{j})}(\tilde{x}a_{K}+b_{K})a_{K}d\tilde{x}$
$\displaystyle\overset{(2)}{=}~{}C\int_{\tfrac{v_{L}-b_{K}}{a_{K}}}^{\tfrac{v_{H}-b_{K}}{a_{K}}}F_{U(K_{j}-1)}(\tilde{x}a_{K}+b_{K})^{1/q-1}K_{j}(F_{V}(\tilde{x}a_{K}+b_{K}))^{K_{j}-1}f_{V}(\tilde{x}a_{K}+b_{K})a_{K}d\tilde{x}$
$\displaystyle\overset{(3)}{\leq}2C\int_{\tfrac{v_{L}-b_{K}}{a_{K}}}^{\tfrac{v_{H}-b_{K}}{a_{K}}}F_{U(K_{j}-1)}(\tilde{x}a_{K}+b_{K})^{1/q-1}(K_{j}-1)(F_{V}(\tilde{x}a_{K}+b_{K}))^{K_{j}-2}f_{V}(\tilde{x}a_{K}+b_{K})a_{K}d\tilde{x}$
$\displaystyle\overset{(4)}{=}2C\int_{\tfrac{v_{L}-b_{K}}{a_{K}}}^{\tfrac{v_{H}-b_{K}}{a_{K}}}F_{U(K_{j}-1)}(\tilde{x}a_{K}+b_{K})^{1/q-1}f_{U(K_{j}-1)}(\tilde{x}a_{K}+b_{K})a_{K}d\tilde{x}$
$\displaystyle\overset{(5)}{=}2C\int_{0}^{1}y^{1/q-1}dy~{}=~{}2C~{}<~{}\infty,$

as desired, where (1) holds by (A.23) and (A.28), (2) and (4) by (A.20), (3)
by $F_{V}(\tilde{x}a_{K}+b_{K})\leq 1$ and $K_{j}\geq K\geq 2$, and (5) by the
change of variables $y=F_{U(K_{j}-1)}(\tilde{x}a_{K}+b_{K})$.

######  Lemma A.3.

For any $h\in\Sigma$, assume that

$\Big{\\{}y\in\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}\int_{\Xi}\kappa_{\xi}(h)f_{\tilde{\mathbf{Z}}|\xi}(h)dW(\xi)\leq\int_{\Xi}f_{(Y_{\mu},\tilde{\mathbf{Z}})|\xi}(y,h)d\Lambda(\xi)\Big{\\}}~{}=~{}[A(h),B(h)],$
(A.29)

where $B(h)-A(h)\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to\mathbb{R}_{+}$
is continuous. Then,

$\tilde{U}(h)~{}=~{}\Big{\\{}y\in\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}\int_{\Xi}\kappa_{\xi}(h)f_{\tilde{\mathbf{Z}}|\xi}(h)dW(\xi)\leq\int_{\Xi}f_{(Y_{\mu},\tilde{\mathbf{Z}})|\xi}(y,h)d\Lambda(\xi)\Big{\\}}.$

###### Proof.

It suffices to show that $[A(h),B(h)]$ satisfies conditions (a)-(c) of Theorem
3.1.

Part (a). Consider the following derivation.

$\displaystyle
P(\partial\\{(y,h)\in\mathbb{R}\times\Sigma\mathrel{\mathop{\mathchar
58\relax}}y\in\tilde{U}(h)\\})$ $\displaystyle~{}\overset{(1)}{\leq}~{}\int
P_{\xi}(Y_{\mu}\in\\{A(h),B(h)\\}|\mathbf{\tilde{Z}}=h)f_{\tilde{\mathbf{Z}}|\xi}(h)dh~{}\overset{(2)}{=}~{}0,$

where (1) holds by (A.29), which implies that
$\partial\\{(y,h)\in\mathbb{R}\times\Sigma\mathrel{\mathop{\mathchar
58\relax}}y\in\tilde{U}(h)\\}~{}\subseteq~{}\\{(A(h),h),(B(h),h)\mathrel{\mathop{\mathchar
58\relax}}h\in\Sigma\\}$, and (2) because
$\\{Y_{\mu}|\mathbf{\tilde{Z}}=h;\xi\\}$ is continuously distributed.

Part (b). Fix $h\in\Sigma$ arbitrarily. Since

$\lim_{|y|\to\infty}\int_{\Xi}f_{(Y_{\mu},\tilde{\mathbf{Z}})|\xi}(y,h)d\Lambda(\xi)=0<\int_{\Xi}\kappa_{\xi}(h)f_{\tilde{\mathbf{Z}}|\xi}(h)dW(\xi),$

$\exists y(h)\in(0,\infty)$ such that $\max\\{|B(h)|,|A(h)|\\}\leq y(h)$ for
all $|y|>y(h)$. Then, $[A(h),B(h)]\subseteq[-y(h),y(h)],$ and so
$\lg([A(h),B(h)])\leq 2y(h)<\infty$.

Part (c). Under (A.29), $\lg([A(h),B(h)])=B(h)-A(h)$ which is assumed
continuous.

######  Lemma A.4.

Assume that $\xi<1$. Let $Z_{1}=H_{\xi}(E_{1})$ and
$Z_{2}=H_{\xi}(E_{1}+E_{2})$ with $E_{1},E_{2}$ i.i.d. standard exponential
random variables and $H_{\xi}$ as in (2.4). Then,

$E[Z_{1}-Z_{2}]~{}=~{}\Gamma(1-\xi),$

where $\Gamma$ is the standard Gamma function.

###### Proof.

The support of $(Z_{1},Z_{2})$ is
$S_{\xi}=\\{(x_{1},x_{2})\mathrel{\mathop{\mathchar 58\relax}}x_{1}\geq
x_{2},~{}\xi x_{1}\geq-1,~{}\xi x_{2}\geq-1\\}$. For any $(x_{1},x_{2})\in
S_{\xi}$, the PDF of $(Z_{1},Z_{2})$ is

$f_{Z_{1},Z_{2}|\xi}(x_{1},x_{2})~{}=~{}\left\\{\begin{array}[]{lc}(1+\xi
x_{1})^{-1/\xi-1}(1+\xi x_{2})^{-1/\xi-1}\exp(-(1+\xi
x_{2})^{-1/\xi})&\text{if }\xi\neq 0,\\\
\exp({-x_{1}})\exp({-x_{2}})\exp({-\exp({-x_{2})})}&\text{if
}\xi=0,\end{array}\right.$

The desired result follows from this formula. We only show the result for
$\xi<0$, as the result for the other two cases is analogous.

$\displaystyle E[Z_{1}-Z_{2}]$
$\displaystyle=~{}\int_{-\infty}^{-1/\xi}\int_{x_{2}}^{-1/\xi}(x_{1}-x_{2})(1+\xi
x_{1})^{-1/\xi-1}(1+\xi x_{2})^{-1/\xi-1}\exp(-(1+\xi
x_{2})^{-1/\xi})dx_{1}dx_{2}$
$\displaystyle\overset{(1)}{=}~{}\frac{1}{\xi^{3}}\int_{\infty}^{0}\int_{t_{2}}^{0}(t_{1}-t_{2})t_{1}^{-1/\xi-1}t_{2}^{-1/\xi-1}\exp(-t_{2}^{-1/\xi})dt_{1}dt_{2}$
$\displaystyle\overset{(2)}{=}~{}\int_{0}^{\infty}(\frac{1}{(1-\xi)})\exp(-v)v^{-\xi+1}dv$
$\displaystyle\overset{(3)}{=}~{}\Gamma(1-\xi),$

as desired, where (1) holds by the change of variables $t_{1}=1+\xi x_{1}$ and
$t_{2}=1+\xi x_{2}$, (2) by the change of variables $v=t_{2}^{-1/\xi}$, and
(3) by $(1-\xi)\Gamma(1-\xi)=\Gamma(2-\xi)$.

######  Lemma A.5.

Assume that $\xi<1$. Let $Z=H_{\xi}(E_{1}+E_{2})$ with $E_{1},E_{2}$ i.i.d.
standard exponential random variables and $H_{\xi}$ as in (2.4). Then,

$E[Z]~{}=~{}\bigg{\\{}\begin{array}[]{ll}(\Gamma(2-\xi)-1)/\xi&\text{if
}\xi\not=0\\\ \bar{\gamma}-1&\text{if }\xi=0\end{array}$

where $\Gamma$ is the Gamma function and $\bar{\gamma}\approx 0.577$ is the
Euler constant.

###### Proof.

Note that the PDF of $Z$ is

$f_{Z|\xi}(z)~{}=~{}\bigg{\\{}\begin{array}[]{ll}(1+\xi
z)^{-{(2+\xi)}/{\xi}}\exp(-(1+\xi z)^{-1/\xi})1[1+\xi z\geq 0]&\text{if
}\xi\neq 0\\\ \exp(-2z)\exp(-\exp(-z))&\text{if }\xi=0.\end{array}$

We only show the result for $\xi\not=0$, as the result for $\xi=0$ is
analogous.

$\displaystyle E[Z]$ $\displaystyle~{}=~{}\int_{-\infty}^{\infty}z(1+\xi
z)^{-\tfrac{2+\xi}{\xi}}\exp(-(1+\xi z)^{-1/\xi})1[1+\xi z\geq 0]dz$
$\displaystyle~{}\overset{(1)}{=}~{}\big{(}\int_{0}^{\infty}(t^{1-\xi}-t)\exp(-t)dt\big{)}/\xi$
$\displaystyle~{}\overset{(2)}{=}~{}(\Gamma(2-\xi)-1)/\xi,$

where (1) holds by the change of variables $u=(1+\xi z)^{-1/\xi}$ and (2) by
$\Gamma(0)=1$.

######  Lemma A.6.

Assume (2.1) holds. Also, assume that $\xi<1$ and
$E[|V|^{1+\varepsilon}]<\infty$ for some $\varepsilon>0$. For any sequence
$\\{x_{K}\mathrel{\mathop{\mathchar 58\relax}}K\in\mathbb{N}\\}$ with
$x_{K}\to x\in\bar{S}_{\xi}\equiv\\{s\mathrel{\mathop{\mathchar
58\relax}}G_{\xi}(s)>0\\}$, $\lim_{K\to\infty}L_{K}(x_{K})=L(x)$, where

$\displaystyle
L_{K}(x)~{}\equiv~{}x-\tfrac{\int_{{(v_{L}-b_{K})/a_{K}}}^{x}{F_{V}(ha_{K}+b_{K})^{K}}dh}{F_{V}(xa_{K}+b_{K})^{K}}~{}~{}~{}\text{and}~{}~{}~{}L(x)~{}\equiv~{}x-\tfrac{\int_{-\infty}^{x}G_{\xi}(h)dh}{G_{\xi}(x)}.$
(A.30)

###### Proof.

Throughout this proof, it is relevant to note that ${F_{V}(ha_{K}+b_{K})^{K}}$
is the CDF of $(V_{(1)}-b_{K})/a_{K}$ and $G_{\xi}$ is the CDF of
$H_{\xi}(E_{1})$, where $V_{(1)}$ denotes the sample maximum of $K$ random
draws from $F_{V}$, $H_{\xi}$ and $E_{1}$ are as in Lemma 2.1.

As a preliminary step, we show that for any $x\in\mathbb{R}$,

$\lim_{K\to\infty}\int_{-\infty}^{x}({F_{V}(ha_{K}+b_{K})^{K}}-G_{\xi}(h))dh~{}=~{}0.$
(A.31)

To this end, consider the following argument for any $K$.

$\displaystyle\int_{-\infty}^{x}({F_{V}(ha_{K}+b_{K})^{K}}-G_{\xi}(h))dh~{}$
$\displaystyle\overset{(1)}{=}\bigg{[}\begin{array}[]{c}x({F_{V}(xa_{K}+b_{K})^{K}}-G_{\xi}(x))+\\\
{{E[H_{\xi}(E_{1})1}[{H_{\xi}(E_{1})\leq
x}]]}-{{E[\tfrac{V_{(1)}-b_{K}}{a_{K}}1}[\tfrac{V_{(1)}-b_{K}}{a_{K}}\leq
x]]}\end{array}\bigg{]}.$ (A.34)

where (1) holds by
$\lim_{h\to-\infty}{F_{V}(ha_{K}+b_{K})^{K}}=\lim_{h\to-\infty}G_{\xi}(h)=0$
and integration by parts.

Lemma 2.1 implies that as $K\to\infty$,

$\tfrac{{V_{(1)}-b_{K}}}{{a_{K}}}~{}\overset{d}{\to}~{}H_{\xi}(E_{1}).$ (A.35)

Since $H_{\xi}(E_{1})$ is continuously distributed, (A.35) implies that

$\lim_{K\to\infty}x({F_{V}(xa_{K}+b_{K})^{K}}-G_{\xi}(x))=0$ (A.36)

Also, note that (A.35) and the continuous mapping theorem imply that as
$K\to\infty$,

${\tfrac{V_{(1)}-b_{K}}{a_{K}}1}[\tfrac{V_{(1)}-b_{K}}{a_{K}}\leq
x]~{}\overset{d}{\to}~{}{H_{\xi}(E_{1})1}[{H_{\xi}(E_{1})\leq x}].$ (A.37)

Let $\delta>0$ and $\bar{K}\in\mathbb{N}$ be as in Lemma A.2. Then, for all
$K\geq\bar{K}$,

$E[|{\tfrac{V_{(1)}-b_{K}}{a_{K}}1}[\tfrac{V_{(1)}-b_{K}}{a_{K}}\leq
x]|^{1+\delta}]~{}\leq~{}E[|\tfrac{{V_{(1)}-b_{K}}}{{a_{K}}}|^{1+\delta}]~{}<~{}\infty.$
(A.38)

Then, (A.37) and (A.38) imply that

$\lim_{K\to\infty}{E[\tfrac{V_{(1)}-b_{K}}{a_{K}}1}[\tfrac{V_{(1)}-b_{K}}{a_{K}}\leq
x]]~{}=~{}{E[H_{\xi}(E_{1})1}[{H_{\xi}(E_{1})\leq x}]].$ (A.39)

Finally, note that (A.31) follows from (A.34), (A.36), and (A.39).

As a second preliminary step, we note that

$\int_{-\infty}^{x}G_{\xi}(h)dh~{}\overset{(1)}{=}~{}xG_{\xi}(x)-\int_{-\infty}^{x}hg_{\xi}(h)dh~{}\leq~{}xG_{\xi}(x)+E[|H_{\xi}(E_{1})|]~{}\overset{(2)}{<}~{}\infty,$
(A.40)

where (1) holds by integration by parts and $\lim_{x\to-\infty}xG_{\xi}(x)=0$,
and (2) by $\xi<1$.

We are now ready to show the desired results. For any $x\in\bar{S}_{\xi}$,
consider the following argument:

$\displaystyle|L_{K}(x_{K})-L(x)|$
$\displaystyle\overset{(1)}{=}~{}\bigg{|}\tfrac{\int_{-\infty}^{x_{K}}{F_{V}(ha_{K}+b_{K})^{K}}dh}{F_{V}(x_{K}a_{K}+b_{K})^{K}}-\tfrac{\int_{-\infty}^{x}G_{\xi}(h)dh}{G_{\xi}(x)}\bigg{|}$
$\displaystyle\overset{(2)}{\leq}~{}\tfrac{\Bigg{|}\begin{array}[]{c}|x_{K}-x|+|\int_{-\infty}^{x}({F_{V}(ha_{K}+b_{K})^{K}}-G_{\xi}(h))dh|\\\
+(\int_{-\infty}^{x}G_{\xi}(h)dh)\sup_{y\in\mathbb{R}}|F_{V}(ya_{K}+b_{K})^{K}-G_{\xi}(y)|\\\
+(\int_{-\infty}^{x}G_{\xi}(h)dh)|G_{\xi}(x_{K})-G_{\xi}(x)|\end{array}\Bigg{|}}{G_{\xi}(x)[G_{\xi}(x)-\sup_{y\in\mathbb{R}}|F_{V}(ya_{K}+b_{K})^{K}-G_{\xi}(y)|-|G_{\xi}(x_{K})-G_{\xi}(x)|]},$
(A.44)

where (1) holds because $F_{V}(v)=0$ for $v<v_{L}$, and (2) by
$\sup_{y\in\mathbb{R}}{F_{V}(ya_{K}+b_{K})^{K}\leq 1}$ and $G_{\xi}(x)\leq 1$.
As $K\to\infty$, we can show that the numerator and the denominator of the
right-hand side of (A.44) converge to zero and $G_{\xi}(x)^{2}>0$,
respectively. This conclusion relies on $x\in\bar{S}_{\xi}$ (and so
$G_{\xi}(x)^{2}>0$), $x_{K}\to x\in\bar{S}_{\xi}$ as $K\to\infty$, the
continuity of $G_{\xi}$, (A.31), (A.40), and van der Vaart (1998, Lemma 2.11).
From this and (A.44), the desired result follows.

######  Lemma A.7.

Assume (2.1) holds. Assume that $\xi<1$ and that
$E[|V_{i,j}|^{1+\varepsilon}]<\infty$ for all $i=1\dots,K_{j}$ in auction
$j=1,\dots,N$ for some $\varepsilon>0$. For any $n\in\mathbb{N}$ and as
$K\to\infty$,

$\big{\\{}\tfrac{P_{j}-b_{K}}{a_{K}}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,n\big{\\}}~{}\overset{d}{\to}~{}\big{\\{}H_{\xi}(E_{1,j})-\tfrac{\int_{-\infty}^{H_{\xi}(E_{1,j})}G_{\xi}(h)dh}{G_{\xi}(H_{\xi}(E_{1,j}))}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,n\big{\\}},$

with
$\\{(a_{K},b_{K})\in\mathbb{R}_{++}\times\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}K\in\mathbb{N}\\}$, $\\{E_{1,j}\mathrel{\mathop{\mathchar
58\relax}}~{}j=1,\dots,n\\}$, and $H_{\xi}$ as in Lemma 2.1.

###### Proof.

Since auctions are independent, it suffices to prove that for any auction
$j=1,\ldots n$, as $K\to\infty$,

$\tfrac{P_{j}-b_{K}}{a_{K}}~{}\overset{d}{\to}~{}H_{\xi}(E_{1,j})-\tfrac{\int_{-\infty}^{H_{\xi}(E_{1,j})}G_{\xi}(h)dh}{G_{\xi}(H_{\xi}(E_{1,j}))},$
(A.45)

where $E_{1,j}$ is a standard exponential random variable. To this end,
consider the following argument.

$\displaystyle\tfrac{P_{j}-b_{K}}{a_{K}}$
$\displaystyle=~{}\big{(}\tfrac{V_{(1),j}-b_{K_{j}-1}}{a_{K_{j}-1}}-\tfrac{(\int_{{v_{L}}}^{V_{(1),j}}{F_{V}(u)^{K_{j}-1}}du)/a_{K_{j}-1}}{F_{V}(V_{(1),j})^{K_{j}-1}}\big{)}\tfrac{a_{K_{j}-1}}{a_{K}}+\tfrac{b_{K_{j}-1}-b_{K}}{a_{K}}$
$\displaystyle\overset{(1)}{=}~{}\big{(}L_{K_{j}-1}\big{(}\tfrac{V_{(1),j}-b_{K_{j}-1}}{a_{K_{j}-1}}\big{)}\big{)}\tfrac{a_{K_{j}-1}}{a_{K}}+\tfrac{b_{K_{j}-1}-b_{K}}{a_{K}},$
(A.46)

where (1) holds by the change of variables $u=ha_{K_{j}-1}+b_{K_{j}-1}$ and
$L_{K}$ as in (A.30).

Since $(K_{j}-1)/K\to 1$ as $K\to\infty$, the same argument as Lemma 2.1
implies that

$\lim_{K\to\infty}\big{(}\tfrac{a_{K_{j}-1}}{a_{K}},\tfrac{b_{K_{j}-1}-b_{K}}{a_{K}}\big{)}~{}=~{}(1,0).$
(A.47)

Next, notice that Lemma 2.1 and (A.47) imply that as $K\to\infty$,

$\tfrac{V_{(1),j}-b_{K_{j}-1}}{a_{K_{j}-1}}~{}=~{}\tfrac{V_{(1),j}-b_{K}}{a_{K}}\tfrac{a_{K}}{a_{K_{j}-1}}+\tfrac{b_{K}-b_{K_{j}-1}}{a_{K}}~{}\overset{d}{\to}~{}H_{\xi}(E_{1}).$
(A.48)

In addition,
$H_{\xi}(E_{1})\in\bar{S}_{\xi}\equiv\\{x\in\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}G_{\xi}(x)>0\\}$. By (A.48), Lemma A.6, and the extended continuous
mapping theorem (e.g. van der Vaart, 1998, Theorem 1.11.1), as $K\to\infty$,

$L_{K_{j}-1}\big{(}\tfrac{V_{(1),j}-b_{K_{j}-1}}{a_{K_{j}-1}}\big{)}~{}\overset{d}{\to}~{}L(H_{\xi}(E_{1}))~{}=~{}O_{p}(1),$
(A.49)

where $L$ is as in (A.30). Then, (A.45) follows from (A.46), (A.47), and
(A.49).

######  Theorem A.1.

Assume (2.1) holds, and that for some $\varepsilon>0$ with
$(1+\varepsilon)\xi<1$, $E[|V_{i,j}|^{1+\varepsilon}]<\infty$ for all
$i=1\dots,K_{j}$ in auction $j=1,\dots,N$. Finally, assume that the CI for
$\mu_{K}$, $U(\mathbf{P})$, is as in (4.9) with
$\tilde{U}\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\mathcal{P}(\mathbb{R})$ that satisfies the following
conditions:

  1. (a)

$P_{\xi}(\\{Y_{\mu},\mathbf{\tilde{X}}\\}\in\partial\\{(y,h)\in\mathbb{R}\times\Sigma\mathrel{\mathop{\mathchar
58\relax}}y\in\tilde{U}(h)\\})=0$, where $\partial A$ denotes the boundary of
$A$.

  2. (b)

$\lg(\tilde{U}(h))<\infty$ for any $h\in\Sigma$, where $\lg(A)$ denotes the
length of $A$ (i.e., $\lg(A)\equiv\int\mathbf{1}[y\in A]dy$).

  3. (c)

For any sequence $\\{h_{\ell}\in\Sigma\\}_{\ell\in\mathbb{N}}$ with
$h_{\ell}\to h\in\Sigma$, $\lg(\tilde{U}(h_{\ell}))\to\lg(\tilde{U}(h))$.

Then, as $K\to\infty$,

  1. 1.

$P(\mu_{K}\in
U(\mathbf{P}))~{}\to~{}P_{\xi}(Y_{\mu}\in\tilde{U}(\mathbf{\tilde{X}}))$,

  2. 2.

$E[\lg(U(\mathbf{P}))]/a_{K}~{}\to~{}E_{\xi}[\kappa_{\xi}(\mathbf{\tilde{X}})\lg(\tilde{U}(\mathbf{\tilde{X}}))]$.

###### Proof.

This proof follows from the same arguments used to prove Theorem 3.1. The main
difference between the proofs is that we replace
$\\{(P_{j}-b_{K})/a_{K}\mathrel{\mathop{\mathchar 58\relax}}j=1,\dots,N\\}$
with $\\{L_{K_{j}}((V_{(1),j}-b_{K})/a_{K})\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,N\\}$ with $L_{K_{j}}$ as in (A.30) (instead of
$\\{(V_{(2),j}-b_{K})/a_{K}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,N\\}$ as in second-price auctions). Then, the vectors
$\\{Z_{j}\mathrel{\mathop{\mathchar 58\relax}}j=1,\dots,n\\}$ and $\tilde{\bf
Z}=\\{\tilde{Z}_{j}\mathrel{\mathop{\mathchar 58\relax}}j=1,\dots,N\\}$ are
replaced by $\\{X_{j}\mathrel{\mathop{\mathchar 58\relax}}j=1,\dots,N\\}$ and
$\mathbf{\tilde{X}}=\\{\tilde{X}_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,N\\}$, as in (4.4) and (4.7), respectively. With these
changes in place, the results follow from repeating the steps used to prove
Theorem 3.1.

######  Theorem A.2.

Assume (2.1) holds, and that for some $\varepsilon>0$ with
$(1+\varepsilon)\xi<1$, $E[|V_{i,j}|^{1+\varepsilon}]<\infty$ for all
$i=1\dots,K_{j}$ in auction $j=1,\dots,N$. Finally, assume that the CI for
$\pi_{K}$, $U(\mathbf{P})$, is as in (4.11) with
$\tilde{U}\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\mathcal{P}(\mathbb{R})$ that satisfies the following
conditions:

  1. (a)

$P_{\xi}(\\{Y_{\pi},\mathbf{\tilde{X}}\\}\in\partial\\{(y,h)\in\mathbb{R}\times\Sigma\mathrel{\mathop{\mathchar
58\relax}}y\in\tilde{U}(h)\\})=0$, where $\partial A$ denotes the boundary of
$A$.

  2. (b)

$\lg(\tilde{U}(h))<\infty$ for any $h\in\Sigma$, where $\lg(A)$ denotes the
length of $A$.

  3. (c)

For any sequence $\\{h_{\ell}\in\Sigma\\}_{\ell\in\mathbb{N}}$ with
$h_{\ell}\to h\in\Sigma$, $\lg(\tilde{U}(h_{\ell}))\to\lg(\tilde{U}(h))$.

Then, as $K\to\infty$,

  1. 1.

$P(\pi_{K}\in
U(\mathbf{P}))~{}\to~{}P_{\xi}(Y_{\pi}\in\tilde{U}(\mathbf{\tilde{X}}))$,

  2. 2.

$E[\lg(U(\mathbf{P}))]/a_{K}~{}\to~{}E_{\xi}[\kappa_{\xi}(\mathbf{\tilde{X}})\lg(\tilde{U}(\mathbf{\tilde{X}}))]$.

###### Proof.

This proof follows from the same arguments as in Theorems 3.1 and A.1.

######  Theorem A.3.

Assume (2.1) holds. In the hypothesis testing problem in (4.12), the test
defined by (4.13) satisfies the following properties:

  1. 1.

It is asymptotically valid and level $\alpha$, i.e.,
$\lim_{K\to\infty}E_{\xi_{0}}[\varphi_{K}^{\ast}(\mathbf{P})]~{}=~{}\alpha.$

  2. 2.

It is asymptotically efficient.

###### Proof.

This proof follows from the same arguments as in Theorems 3.3 and 3.4.

###  A.3 Computational details

####  A.3.1 Second-price auctions

This section provides computational details for objects introduced in Section
3. Throughout this section, we use
$\mathbf{\tilde{z}}=(z_{1},\dots,z_{N})\in\Sigma$ and $N=n-2$.

First, recall that $f_{\mathbf{\tilde{Z}}|\xi}(\mathbf{\tilde{z}})$ is as
computed in (3.15). For $\mathbf{\tilde{z}}\in\Sigma$, we can compute:

$\kappa_{\xi}(\tilde{\mathbf{z}})f_{\tilde{\mathbf{Z}}|\xi}(\tilde{\mathbf{z}})=n!(\Gamma(2n-\xi))\int_{0}^{b(\xi)}s^{n-1}\exp\bigg{(}\begin{array}[]{c}(\xi-2n)\log\big{(}\sum_{j=1}^{n}(1+z_{j}\xi
s)^{-1/\xi}\big{)}\\\ -(1+\tfrac{2}{\xi})\big{(}\sum_{j=1}^{n}\log(1+z_{j}\xi
s)\big{)}\end{array}\bigg{)}ds,$ (A.50)

where
$\kappa_{\xi}(\mathbf{\tilde{z}})=E[Z_{(n)}-Z_{(1)}|\mathbf{\tilde{Z}}=\mathbf{\tilde{z}}]$
and $b(\xi)=-1/\xi$ for $\xi<0$, and $b(\xi)=\infty$ otherwise. For
$(y,\mathbf{\tilde{z}})\in\mathbb{R}_{+}\times\Sigma$, calculations yield:

$\displaystyle f_{Y_{\mu},\tilde{\mathbf{Z}}|\xi}(y,\tilde{\mathbf{z}})=$
$\displaystyle\tfrac{n!(\Gamma(1-\xi))^{n-1}}{y^{n}}\int_{a_{\mu}(\xi)}^{c_{\mu}(\xi)}\exp\bigg{(}\begin{array}[]{c}-\sum_{j=1}^{n}(1+\xi
s+(\Gamma(1-\xi))\xi{z_{j}}/{y})^{-1/\xi}\\\
-(1+\tfrac{2}{\xi})\sum_{j=1}^{n}\log(1+\xi s+(\Gamma(1-\xi))\xi
z_{j}/{y})\end{array}\bigg{)}ds,$ (A.53)

where $a_{\mu}(\xi)$ and $c_{\mu}(\xi)$ are defined such that for all
$s\in(a_{\mu}(\xi),c_{\mu}(\xi))$, we have $1+\xi s+(\Gamma(1-\xi))\xi/y>0$
and $1+\xi s>0$. For $(y,\mathbf{\tilde{z}})\in\mathbb{R}\times\Sigma$, more
calculations yield:

$\displaystyle f_{Y_{\pi},\tilde{\mathbf{Z}}|\xi}(y,\tilde{\mathbf{z}})=$
$\displaystyle\tfrac{n!}{|y|^{n}}\int_{a_{\pi}(\xi)}^{c_{\pi}(\xi)}{|\pi-s|}^{n-1}\exp\bigg{(}\begin{array}[]{c}-\sum_{j=1}^{n}(1+\xi(s+{z_{j}}(\pi-s)/y)^{-1/\xi}\\\
-(1+\tfrac{2}{\xi})\sum_{j=1}^{n}\log(1+\xi(s+z_{j}(\pi-s)/y)\end{array}\bigg{)}ds,$
(A.56)

where $a_{\mu}(\xi)$ and $c_{\mu}(\xi)$ are such that for all
$s\in(a_{\mu}(\xi),c_{\mu}(\xi))$, we have $1+\xi(s+(\pi-s)/y>0$, $1+\xi s>0$,
and $(\pi-s)/y>0$. The integrals in (A.50), (A.53), and (A.56) can be
approximated by Gaussian quadrature.

The CI’s in Section 3 require Lagrange multipliers $\Lambda$ that we compute
using the algorithm developed by Elliott et al. (2015). See Müller and Wang
(2017) for another application of this algorithm. We now provide a detailed
description of how we implemented this algorithm:

  1. 1)

Discretize $\Xi$ into a fine grid
$\Xi_{M}\equiv\\{\xi_{1},\xi_{2},\ldots,\xi_{M}\\}$ between
$\xi_{1}=\inf\\{\Xi\\}$ and $\xi_{M+1}=\sup\\{\Xi\\}$ (we use $M=50$ uniformly
located points between $\xi_{1}$ and $\xi_{M+1}$). Set $s=1$, and define an
arbitrary set of initial positive weights
$\lambda^{(s)}=\\{\lambda^{(s)}_{m}\mathrel{\mathop{\mathchar
58\relax}}m=1,\dots,M\\}$ over $\Xi_{M}$ (we use a uniform weights, i.e.,
$\lambda^{(1)}=\\{1/M,\ldots,1/M\\}$).

  2. 2)

For each $m=1,\dots,M$, simulate a large number $B$ of $n$ i.i.d. draws the EV
distribution with parameter $\xi_{m}$ (we use $B=10,000$). For each
$m=1,\dots,M$ and $b=1,\dots,B$, the samples is denoted by
$\mathbf{Z}_{\xi_{m}}(b)=\\{Z_{\xi_{m},1}(b),\dots,Z_{\xi_{m},n}(b)\\}$. By
applying (3.18) to each sample, we get $N=n-2$ sorted and normalized draws,
denoted by
$\tilde{\mathbf{Z}}_{\xi_{m}}(b)=\\{\tilde{Z}_{\xi_{m},1}(b),\dots,\tilde{Z}_{\xi_{m},N}(b)\\}\in\Sigma$.

  3. 3)

For each $m=1,\dots,M$, use the random draws in step 2) to approximate the
limiting coverage probabilities for parameter $\xi_{m}$ in the following
manner:

    1. (a)

For the winner’s expected utility, we approximate
${P}_{\xi_{m}}({\Gamma(1-\xi_{m})}/({Z_{(n)}-Z_{(1)}})\in
U(\tilde{\mathbf{Z}}))$ with

$\mathbb{\hat{P}}_{m}~{}\equiv~{}\tfrac{1}{B}\sum_{b=1}^{B}1\big{(}\tfrac{\Gamma(1-\xi_{m})}{Z_{\xi_{m},(1)}(b)-Z_{\xi_{m},(n)}(b)}\in
U(\tilde{\mathbf{Z}}_{\xi_{m}}(b))\big{)},$

where $\Gamma$ is the standard Gamma function and
$U(\tilde{\mathbf{Z}}_{\xi_{m}}(b))$ is as in (3.23), involving integrals of
(A.50) and (A.53).

    2. (b)

For the seller’s expected revenue, we approximate
${P}_{\xi_{m}}(({\pi(\xi_{m})-Z_{(1)}})/({Z_{(n)}-Z_{(1)}})\in
U(\tilde{\mathbf{Z}})))$, with

$\mathbb{\hat{P}}_{m}~{}\equiv~{}\tfrac{1}{B}\sum_{b=1}^{B}1\big{(}\tfrac{\pi(\xi_{m})-Z_{\xi_{m},(n)}(b)}{Z_{\xi_{m},(1)}(b)-Z_{\xi_{m},(n)}(b)}\in
U(\tilde{\mathbf{Z}}_{\xi_{m}}(b))\big{)},$

where $\pi(\xi_{m})=(\Gamma(2-\xi_{m})-1)/\xi_{m}$ with $\Gamma$ is the
standard Gamma function if $\xi_{m}\not=0$, and $\pi(\xi_{m})=-1+\bar{\gamma}$
if $\xi_{m}=0$ with $\bar{\gamma}\approx 0.577$ equal to the Euler’s constant,
and $U(\tilde{\mathbf{Z}}_{\xi_{m}}(b))$ is as in (3.28), involving integrals
of (A.50) and (A.53).

  4. 4)

Update the weights by setting
$\lambda_{m}^{(s+1)}=\lambda_{m}^{(s)}+\varepsilon((1-\mathbb{\hat{P}}_{m})-\alpha)$
for all $m=1,\dots,M$, where $\alpha$ is the significance level and
$\varepsilon>0$ is a small step length (we use $\varepsilon=0.05$).
Intuitively, the weight on $\xi_{m}$ is decreased or increased if the CI has
overcoverage and undercoverage, respectively.

  5. 5)

Repeat steps 3)-4) a large number of times $S$ (we use $S=2,000$), to get
$\lambda^{(S)}=\\{\lambda_{m}^{(S)}\mathrel{\mathop{\mathchar
58\relax}}m=1,\dots,M\\}$. The Lagrange multipliers $\Lambda$ is obtained by
interpolating $\lambda^{(S)}$ from $\Xi_{M}$ to $\Xi$.

If the algorithm’s tuning parameters are appropriately chosen, the Lagrange
multipliers generated by it yield a CI that (a) approximately minimizes
asymptotic weighted length (due to the reliance on (3.23) or (3.28)), and (b)
achieves approximately size control for all $\xi\in\Xi_{M}$, which should
extend to all $\xi\in\Xi$ by continuity. Finally, it is worth noting that
these Lagrange multipliers only need to be determined once for a given value
of $n$ and set $\Xi$. The tables of the Lagrange multipliers and the
corresponding MATLAB code are available on our website.

####  A.3.2 First-price auctions

We now consider analogous expressions for first-price auctions. Let
$j=1,\dots,N$ denote an arbitrary first-price auction. Relative to the
previous section, the main difficulty here is that PDF of $X_{j}$ in (4.4)
does not generally have a closed-form expression and, thus, we do not have an
analog of (3.15) for first-price auctions.

To explain this issue, we now provide an implicit formula for the PDF of
$X_{j}$ for any arbitrary auction $j=1,\dots,N$. Consider the following
argument for $\xi<0$:

$\displaystyle X_{j}$
$\displaystyle~{}\overset{(1)}{=}~{}H_{1}-\tfrac{1}{G_{\xi}(H_{1})}\int_{-\infty}^{H_{1}}G_{\xi}(h)dh$
$\displaystyle~{}\overset{(2)}{=}~{}H_{1}-\exp((1+\xi
H_{1})^{-1/\xi})\int_{-\infty}^{H_{1}}\exp(-(1+\xi h)^{-1/\xi})dh$
$\displaystyle~{}\overset{(3)}{=}~{}H_{1}-\exp((1+\xi
H_{1})^{-1/\xi})\Gamma(-\xi,(1+\xi H_{1})^{-1/\xi})$
$\displaystyle~{}\overset{(4)}{=}~{}[{(E_{1,j})^{-\xi}-1}]/{\xi}-\exp(E_{1,j})\Gamma(-\xi,E_{1,j})$
$\displaystyle~{}\overset{(5)}{=}~{}[{\exp(E_{1,j})\Gamma(1-\xi,E_{1,j})-1}]/{\xi},$
(A.57)

where (1) holds by (4.4) and denoting $H_{1}=H_{\xi}(E_{1,j})$ with
$\\{E_{1,j}\mathrel{\mathop{\mathchar 58\relax}}~{}j=1,\dots,n\\}$ and
$H_{\xi}$ as specified in Lemma 2.1, (2) by (2.2), (3) by the change of
variables $u=(1+\xi h)^{-1/\xi}$ and using
$\Gamma(a,x)=\int_{x}^{\infty}u^{a-1}\exp(-u)du$ to denote the upper
incomplete Gamma function, (4) by $E_{1,j}=H_{\xi}^{-1}(H_{1})$, and (5) by
$\Gamma(1+a,x)=a\Gamma(a,x)+x^{a}\exp(-x)$ applied to $a=-\xi$ and
$x=E_{1,j}$. By a similar derivation for $\xi>0$ and $\xi=0$, we get
$X_{j}~{}=~{}e_{\xi}(E_{1,j})$ with

$e_{\xi}(x)~{}\equiv~{}\bigg{\\{}\begin{array}[]{ll}[\exp(x)\Gamma(1-\xi,x)-1]/\xi&\text{
if }\xi\neq 0,\\\ -\ln(x)-\exp(x)\Gamma(0,x)&\text{ if }\xi=0.\end{array}$
(A.58)

We numerically verified that $e_{\xi}(x)$ is decreasing in $x$ for all $\xi$,
and so the PDF of $X_{j}$ can be expressed as follows:

$f_{X_{j}|\xi}(x)~{}=~{}-\tfrac{\partial e_{\xi}^{-1}(x)}{\partial
x}\exp(-e_{\xi}^{-1}(x)).$ (A.59)

We can use (A.59) to derive an implicit expression for the joint PDF of
$\mathbf{\tilde{X}}=\\{\tilde{X}_{j}\mathrel{\mathop{\mathchar
58\relax}}j=1,\dots,N\\}\in\Sigma$ with $\tilde{X}_{j}$ as in (4.7). The main
difficulty relative to the second-price auction is that neither $e_{\xi}$ in
(A.58) nor its inverse have closed-form expressions, and so evaluating them
repeatedly is computationally challenging. For this reason, we do not have a
closed-form analog of (3.15) for first-price auctions.

To deal with the aforementioned computational issues, we propose a numerical
approximation based on a series expansion of the incomplete Gamma function in
Abramowitz and Stegun (1964, Page 263): for $x$ sufficiently large,

$\Gamma(1-\xi,x)~{}\approx~{}x^{-\xi}\exp(-x)[1+\tfrac{(-\xi)}{x}+\tfrac{(-\xi)(-\xi-1)}{x^{2}}+\tfrac{(-\xi)(-\xi-1)(-\xi-2)}{x^{3}}+\dots].$
(A.60)

A first-order approximation based on (A.60) gives
$\ln(\exp(x)\Gamma(1-\xi,x))\approx-\xi\ln x$. To allow for approximation
errors, we propose the following equation:

$\ln(\exp(x)\Gamma(1-\xi,x))~{}\approx~{}r_{2}(\xi)-r_{1}(\xi)\ln x,$ (A.61)

where $r_{1}(\xi)$ and $r_{2}(\xi)$ are functions of $\xi$.555If the first-
order approximation was exact, we would have $r_{1}(\xi)=\xi$ and
$r_{2}(\xi)=0$. To find suitable values for these functions we fit an OLS
regression of $\ln(\Gamma(1-\xi,x)\exp(x))$ on $\ln x$ and a constant for a
grid of 50,000 equally-spaced values of $x$ between $10^{-6}$ and $1-10^{-6}$
quantiles of the exponential distribution. We use the OLS coefficients as our
values for $(-r_{1}(\xi),r_{2}(\xi))$. By combining (A.58) and (A.61), we get

$e_{\xi}(x)\approx\bigg{\\{}\begin{array}[]{ll}[x^{-r_{1}(\xi)}\exp(r_{2}(\xi))-1]/\xi&\text{
if }\xi\neq 0,\\\ (r_{1}(0)-1)\ln x-r_{2}(0)&\text{ if }\xi=0.\end{array}$
(A.62)

By (A.62) and using $r_{3}(\xi)=\exp(r_{2}(\xi)/r_{1}(\xi))$, we get

$\displaystyle e_{\xi}^{-1}(x)$
$\displaystyle\approx\bigg{\\{}\begin{array}[]{ll}r_{3}(\xi)(1+\xi
x)^{-1/r_{1}(\xi)}&\text{ if }\xi\neq 0,\\\
\exp[(x+r_{2}(0))/(r_{1}(0)-1)]&\text{ if }\xi=0,\end{array}$ (A.65)
$\displaystyle\tfrac{\partial e_{\xi}^{-1}(x)}{\partial x}$
$\displaystyle\approx\bigg{\\{}\begin{array}[]{ll}-(1+\xi
x)^{-(1/r_{1}(\xi)+1)}\xi r_{3}(\xi)/r_{1}(\xi)&\text{ if }\xi\neq 0,\\\
\exp[(x+r_{2}(0))/(r_{1}(0)-1)]/(r_{1}(0)-1)&\text{ if }\xi=0.\end{array}$
(A.68)

By combining (A.59) and (A.68), we obtain an approximation of the joint
distribution of $\tilde{\mathbf{X}}$ and related functions. Ignoring the case
$\xi=0$, then for $\mathbf{\tilde{x}}\in\Sigma$ and $N=n-2$ we get

$\displaystyle
f_{\tilde{\mathbf{X}}|\xi}(\tilde{\mathbf{x}})=\tfrac{n!\Gamma(n)|r_{1}(\xi)|}{|\xi|}\int_{0}^{b(\xi)}s^{n-2}\exp\bigg{(}\begin{array}[]{c}-n\ln(r_{3}(\xi)\sum_{j=1}^{n}(1+\xi\tilde{x}_{j}s)^{-1/r_{1}(\xi)})\\\
-(1/r_{1}(\xi)+1)\sum_{j=1}^{n}\ln(1+\xi\tilde{x}_{j}s)\\\ +n\ln(|\xi
r_{3}(\xi)/r_{1}(\xi)|)\end{array}\bigg{)}ds,$ (A.72)

where $b(\xi)=-1/\xi$ for $\xi<0$, and $b(\xi)=\infty$ otherwise. This
expression is the analog of (3.15) for first-price auctions. Analogously, we
have

$\displaystyle\kappa_{\xi}(\tilde{\mathbf{x}})f_{\tilde{\mathbf{X}}|\xi}(\tilde{\mathbf{x}})=$
$\displaystyle\tfrac{n!|r_{1}(\xi)|\Gamma(n-r_{1}(\xi))}{|\xi|}\int_{0}^{b(\xi)}s^{n-1}\exp\Bigg{(}\begin{array}[]{c}(r_{1}(\xi)-n)\ln(r_{3}(\xi)\sum_{j=1}^{n}(1+\xi\tilde{x}_{j}s)^{-1/r_{1}(\xi)})\\\
-(1/r_{1}(\xi)+1)\sum_{j=1}^{n}\ln(1+\xi\tilde{x}_{j}s)\\\ +n\ln(|\xi
r_{3}(\xi)/r_{1}(\xi)|)\end{array}\Bigg{)}ds,$ (A.76)

where
$\kappa_{\xi}(\mathbf{x})=E[X_{(n)}-X_{(1)}|\mathbf{\tilde{X}}=\mathbf{x}]$.
For $(y,\mathbf{\tilde{x}})\in\mathbb{R}_{+}\times\Sigma$, we also have

$\displaystyle f_{Y_{\pi},\tilde{\mathbf{X}}|\xi}(y,\tilde{\mathbf{x}})=$
$\displaystyle\tfrac{1}{|y|^{n}}\int_{a_{\pi}(\xi)}^{c_{\pi}(\xi)}|\pi-s|^{n-1}\exp\Bigg{(}\begin{array}[]{c}-(1/r_{1}(\xi)+1)\sum_{j=1}^{n}\ln(1+\xi(s+\tilde{x}_{j}(\pi-s)/y))\\\
-r_{3}(\xi)\sum_{j=1}^{n}(1+\xi(s+\tilde{x}_{j}(\pi-s)/y))^{-1/r_{1}(\xi)}\\\
+n\ln(|\xi r_{3}(\xi)/r_{1}(\xi)|)\end{array}\Bigg{)}ds,$ (A.80)

where $a_{\pi}(\xi)$ and $c_{\pi}(\xi)$ are such that for all
$s\in(a_{\pi}(\xi),c_{\pi}(\xi))$, we have $1+\xi(s+(\pi-s)/y)>0$, $1+\xi
s>0$, and $(\pi-s)/y>0$. The integrals in (A.72), (A.76), and (A.80) can be
approximated by Gaussian quadrature. Given these expressions, we can use the
algorithm in Section A.3.1 to compute the CI.

###  A.4 Additional Monte Carlo simulations

This section provides Monte Carlo simulations for first-price auctions. We
first consider the problem of inference on the winner’s expected utility using
only transaction prices. The parameters of the simulations are as in Section
5.

We consider three CIs for $\mu_{K}$ that are analogous to those used in
Section 5:

  1. (i)

This is our proposed CI in Section 4.1. As in Section 5, this method is
implemented by constructing $U(\mathbf{P})$ in (4.9) with $\Xi=[-1,0.5]$ and
$W$ equal to the uniform distribution over this interval. As before, the
validity of this CI is based on asymptotics as $K\to\infty$.

  2. (ii)

A CI based on observing the transaction price and the highest valuation for
each auction. These data are infeasible in empirical applications of first-
price auctions (valuations are unobserved), but we take it as a benchmark for
any method that relies on the traditional asymptotics with $n\to\infty$. These
data allows us to compute $D_{j}\equiv V_{(1),j}-P_{j}$ for each auction
$j=1,\dots,n$. In turn, this information enables us to test
$H_{0}\mathrel{\mathop{\mathchar 58\relax}}\mu_{K}=b$ using the t-statistic
$\sqrt{n}(\bar{D}_{n}-b)/s$, where $\bar{D}_{n}=\sum_{j=1}^{n}D_{j}/n$ and
$s^{2}={\sum^{n}_{j=1}(D_{j}-\bar{D}_{n})^{2}/(n-1)}$. Under $H_{0}$, standard
asymptotic arguments imply that the t-statistic converges to a standard normal
distribution as $n\to\infty$. One can then construct a CI by inverting the
aforementioned hypothesis tests. In contrast to the first CI, the validity of
this method relies on standard asymptotic arguments with $n\to\infty$.

  3. (iii)

A bootstrap-based CI based on observing the highest valuations for each
auction. This approach is obviously infeasible in using data from first-price
auctions, but it could be feasible with bid data from second-price auctions;
cf. Menzel and Morganti (2013). Observing the highest valuations across
auctions allows us to identify the distribution of valuations which, in turn,
allows us to identify $\mu_{K}$ (further details are provided in Section A.5).
By replacing identified objects with suitable estimators, one can consistently
estimate $\mu_{K}$. In addition, one can repeat this process using bootstrap
samples to construct a CI for $\mu_{K}$. The method is implemented with 500
bootstrap samples. The validity of this method relies on standard asymptotic
arguments with $n\to\infty$.

Table 4 presents the coverage and length of the three CIs. Our proposed CI
suffers from a slight undercoverage when valuations are $U(0,3)$ or when $n$
is significantly larger than $K$. As our theory predicts, this undercoverage
should diminish as $K$ increases. On the other hand, our proposed CI
demonstrates excellent size control for the non-$U(0,3)$ valuations and $K$ is
significantly larger than $n$. The other two methods perform analogously to
Table 1. The CI based on observing the highest valuation and the transaction
price and the asymptotics $n\to\infty$ are relatively shorter and tend to
suffer from undercoverage, especially when $n$ is small or valuations are
Pareto distributed. In turn, the bootstrap-based CI tends to suffer
significant undercoverage problems.

# Bidders | $K=10$ | $K=100$
---|---|---
# Auctions | $n=10$ | $n=100$ | $n=10$ | $n=100$
Dist. | Cov | Lgth | Cov | Lgth | Cov | Lgth | Cov | Lgth
Method (i): Our proposed CI, asy. w. $K\to\infty$
$U(0,3)$ | 0.89 | 1.20 | 0.89 | 0.29 | 0.91 | 0.22 | 0.98 | 0.12
$|N(0,1)|$ | 0.94 | 1.28 | 0.84 | 0.40 | 0.97 | 1.46 | 0.98 | 0.33
$|t(20)|$ | 0.95 | 1.29 | 0.86 | 0.46 | 0.98 | 1.30 | 0.97 | 0.42
$Pa(0.25)$ | 0.97 | 1.47 | 0.87 | 0.54 | 0.97 | 2.12 | 0.95 | 0.99
Method (ii): CI based on highest valuations, asy. w. $n\to\infty$
$U(0,3)$ | 0.89 | 0.03 | 0.95 | 0.01 | 0.85 | 0.00 | 0.95 | 0.00
$|N(0,1)|$ | 0.86 | 0.31 | 0.94 | 0.11 | 0.86 | 0.24 | 0.93 | 0.09
$|t(20)|$ | 0.88 | 0.40 | 0.93 | 0.14 | 0.83 | 0.37 | 0.92 | 0.13
$Pa(0.25)$ | 0.81 | 0.69 | 0.88 | 0.28 | 0.76 | 1.22 | 0.91 | 0.47
Method (iii): CI based on bootstrap & highest valuations, asy. w. $n\to\infty$
$U(0,3)$ | 0.35 | 0.19 | 0.89 | 0.11 | 0.36 | 0.02 | 0.89 | 0.01
$|N(0,1)|$ | 0.28 | 0.29 | 0.84 | 0.12 | 0.28 | 0.21 | 0.85 | 0.09
$|t(20)|$ | 0.29 | 0.34 | 0.85 | 0.14 | 0.27 | 0.28 | 0.82 | 0.12
$Pa(0.25)$ | 0.20 | 0.34 | 0.69 | 0.20 | 0.23 | 0.61 | 0.69 | 0.35
Table 4: Empirical coverage frequency (Cov) and length (Lgth) of various CIs
for the winner’s expected utility $\mu_{K}$ in first-price auctions. The
results are the average of 500 simulation draws and a nominal coverage level
of 95%.

Finally, we consider the problem of hypothesis testing about the tail index
using only transaction prices. We focus on testing
$H_{0}\mathrel{\mathop{\mathchar 58\relax}}\xi=-1$ vs.
$H_{1}\mathrel{\mathop{\mathchar 58\relax}}\xi\in\Xi/\\{-1\\}$, with
$\Xi=[-1,0.5]$ and $W$ equal to the uniform distribution over this interval.
Table 5 shows the average rejection rate of the test proposed in Section 4.3
over 500 simulations using a desired nominal size of $5\%$. Under $H_{0}$
(i.e., when valuations are $U(0,3)$), our proposed test controls size as long
as the number of bidders is not too small relative to the sample size. Under
$H_{1}$ (i.e., when valuations are $|N(0,1)|$, $|t(20)|$, and $Pa(0.25)$), our
proposed test exhibits adequate power.

# Bidders | 10 |  | 20 |  | 100
---|---|---|---|---|---
# Auctions | 10 | 100 |  | 10 | 100 |  | 10 | 100
$U(0,3)$ | 0.06 | 0.09 |  | 0.04 | 0.06 |  | 0.04 | 0.05
$|N(0,1)|$ | 0.25 | 1.00 |  | 0.22 | 1.00 |  | 0.28 | 1.00
$|t(20)|$ | 0.27 | 1.00 |  | 0.29 | 1.00 |  | 0.34 | 1.00
$Pa(0.25)$ | 0.54 | 1.00 |  | 0.55 | 1.00 |  | 0.55 | 1.00
Table 5: Empirical rejection rate of the test proposed in Section 4.3 for
$H_{0}\mathrel{\mathop{\mathchar 58\relax}}\xi=-1$ in first-price auctions.
The results are the average of 500 simulation draws and a nominal size of 5%.

###  A.5 Auxiliary derivations related to the Monte Carlo simulations

We first provide auxiliary derivations related to the third CI considered in
Section 5. In a second-price auction $j=1,\ldots,n$ with $K$ bidders, the CDF
of the transaction price satisfies:

$F_{P_{j}}(x)~{}\overset{(1)}{=}~{}F_{V_{(2),j}}(x)~{}\overset{(2)}{=}~{}F_{V}(x)^{K}+KF_{V}(x)^{K-1}(1-F_{V}(x)),$
(A.81)

where (1) holds by (3.2) and (2) by the fact that bidders’ valuations are
i.i.d. The previous equation expresses the CDF of the transaction price as a
function of the CDF of valuations. By inverting this mapping, we can derive a
relationship between the distribution of transaction prices and the
distribution of valuations. Furthermore, we have the following connection
between the distribution of valuations and the winner’s expected utility:

$\displaystyle\mu_{K}$ $\displaystyle~{}\overset{(1)}{=}~{}E[V_{(1),j}-P_{j}]$
$\displaystyle~{}\overset{(2)}{=}~{}E[V_{(1),j}-V_{(2),j}]$
$\displaystyle~{}\overset{(3)}{=}~{}K\int_{0}^{\infty}xF_{V}(x)^{K-1}f_{V}(x)dx-(K-1)K\int_{0}^{\infty}x(f_{V}(x)(1-F_{V}(x))F_{V}(x)^{K-2})dx$
$\displaystyle~{}\overset{(4)}{=}~{}K\bigg{(}\int_{0}^{\infty}(1-F_{V}^{K}(x))dx-\int_{0}^{\infty}(1-F_{V}^{K-1}(x))dx\bigg{)},$

where (1) holds because the auctions are i.i.d. (which holds in all of our
Monte Carlo designs), (2) holds by (3.2), (3) by the fact that bidders are
i.i.d and valuations are nonnegative (which holds in all of our Monte Carlo
designs), and (4) by integration by parts.

To conclude, we provide auxiliary derivations related to the third CI
considered in Section A.4. Recall that this method presumes that we observe
the highest valuation $V_{(1),j}$ for each auction $j=1,\dots,n$. Given that
there are $K$ bidders, the CDF of the highest valuation satisfies:

$F_{V_{(1)}}(x)~{}\overset{(1)}{=}~{}F_{V}(x)^{K},$ (A.82)

where (1) holds by the fact that bidders’ valuations are i.i.d. and
$V_{(1)}=\max\\{V_{1},V_{2},\dots,V_{K}\\}$. Furthermore, we can connect the
distribution of valuations and the winner’s expected utility in the following
fashion:

$\displaystyle\mu_{K}~{}$
$\displaystyle\overset{(1)}{=}~{}E\bigg{[}\tfrac{\int_{0}^{V_{(1)}}{F_{V}(u)^{K-1}}du}{F_{V}(V_{(1)})^{K-1}}\bigg{]}$
$\displaystyle\overset{(2)}{=}~{}\int^{\infty}_{0}\bigg{(}\int_{0}^{v}KF_{V}^{K-1}(x)dx\bigg{)}f_{V}(v)dv$
$\displaystyle\overset{(3)}{=}~{}K\int^{\infty}_{0}(1-F_{V}(v))F_{V}(v)^{K-1}dv,$

where (1) holds by (4.8) and by the fact that valuations are nonnegative and
auctions are i.i.d. (which holds in all of our Monte Carlo designs), (2) by
(A.82), and (3) by integration by parts.

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