Tail product-limit process for truncated data with application to extreme
value index estimation

Souad Benchaira, Djamel Meraghni, Abdelhakim Necir∗

Laboratory of Applied Mathematics, Mohamed Khider University, Biskra, Algeria

Abstract

A weighted Gaussian approximation to tail product-limit process for Pareto-
like distributions of randomly right-truncated data is provided and a new
consistent and asymptotically normal estimator of the extreme value index is
derived. A simulation study is carried out to evaluate the finite sample
behavior of the proposed estimator.

Keywords: Empirical process; Extreme value index; Heavy-tails; Hill estimator;
Lynden-Bell estimator; Random truncation.

AMS 2010 Subject Classification: 60G70, 60F17, 62G30.

${}^{\text{*}}$Corresponding author<EMAIL_ADDRESS>

E-mail addresses:
<EMAIL_ADDRESS>(S. Benchaira)
<EMAIL_ADDRESS>(D. Meraghni)

##  1\. Introduction

Let $\left(\mathbf{X}_{i},\mathbf{Y}_{i}\right),$ $1\leq i\leq N$ be a sample
of size $N\geq 1$ from a couple $\left(\mathbf{X},\mathbf{Y}\right)$ of
independent random variables (rv’s) defined over some probability space
$\left(\Omega,\mathcal{A},\mathbf{P}\right),$ with continuous marginal
distribution functions (df’s) $\mathbf{F}$ and $\mathbf{G}$ respectively.
Suppose that $\mathbf{X}$ is truncated to the right by $\mathbf{Y},$ in the
sense that $\mathbf{X}_{i}$ is only observed when
$\mathbf{X}_{i}\leq\mathbf{Y}_{i}.$ This model of randomly truncated data
commonly finds its applications in such areas like astronomy, economics,
medicine and insurance. In the actuarial world, for instance, it is usual that
the insurer claim data do not correspond to the underlying losses, because
they are truncated from above. Indeed, when facing large claims, the insurance
company specifies an upper limit to the amounts to be paid out. The excesses
over this fixed threshold are then covered by a reinsurance company. This kind
of reinsurance is called excess-loss reinsurance (see, e.g., Embrechts et al.,
1997). In life insurance, the upper limit, which may be random, is called the
cedent company retention level whereas in non-life insurance, it is called the
deductible. The usefulness of the statistical analysis under random truncation
is shown in Herbst (1999), where the author applies truncated model techniques
to estimate loss reserves for IBNR (incurred but not reported) claim amounts.
For a recent discussion on randomly right-truncated insurance claims, one
refers to Escudero and Ortega (2008). Some examples of truncated data from
astronomy and economics can be found in Woodroofe (1985) and for applications
in the analysis of AIDS data, see Wang (1989). In reliability, a real dataset,
consisting in lifetimes of automobile brake pads and already considered by
Lawless (2002) in page 69, was recently analyzed in GS2014 as an application
of randomly truncated heavy-tailed models. Since the focus is on datasets that
contain extreme values, then it would be natural to assume that both survival
functions $\overline{\mathbf{F}}:=1-\mathbf{F}$ and
$\overline{\mathbf{G}}:=1-\mathbf{G}$ are regularly varying at infinity with
tail indices $\gamma_{1}>0$ and $\gamma_{2}>0$ respectively. That is, we have,
for any $x>0,$

$\lim_{z\rightarrow\infty}\frac{\overline{\mathbf{F}}\left(xz\right)}{\overline{\mathbf{F}}\left(z\right)}=x^{-1/\gamma_{1}}\text{
and
}\lim_{z\rightarrow\infty}\frac{\overline{\mathbf{G}}\left(xz\right)}{\overline{\mathbf{G}}\left(z\right)}=x^{-1/\gamma_{2}}.$
(1.1)

This class of distributions, which includes models such as Pareto, Burr,
Fréchet, Lévy-stable and log-gamma, takes a prominent role in extreme value
theory. Also known as heavy-tailed, Pareto-type or Pareto-like distributions,
they provide appropriate descriptions for large insurance claims, log-returns,
large price fluctuations, etc… (see, e.g., Resnick, 2006).

Let us denote $\left(X_{i},Y_{i}\right),$ $i=1,...,n$ to be the observed data,
as copies of a couple of rv’s $\left(X,Y\right),$ corresponding to the
truncated sample $\left(\mathbf{X}_{i},\mathbf{Y}_{i}\right),$ $i=1,...,N,$
where $n=n_{N}$ is a sequence of discrete rv’s. By of the law of large
numbers, we have
$n/N\overset{\mathbf{P}}{\rightarrow}p:=\mathbf{P}\left(\mathbf{X}\leq\mathbf{Y}\right)$
as$\ n\overset{\mathbf{P}}{\rightarrow}\infty.$ For convenience, throughout
the paper, the convergence in probability of $n$ and/or any of its
subsequences is simply denoted by $\rightarrow$ instead of
$\overset{\mathbf{P}}{\rightarrow}.$ The joint distribution of $X_{i}$ and
$Y_{i}$ is

$\begin{array}[c]{ll}H\left(x,y\right)&:=\mathbf{P}\left(X\leq x,Y\leq
y\right)\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
&=\mathbf{P}\left(\mathbf{X}\leq x,\mathbf{Y}\leq
y\mid\mathbf{X}\leq\mathbf{Y}\right)=p^{-1}{\displaystyle\int_{0}^{y}}\mathbf{F}\left(\min\left(x,z\right)\right)d\mathbf{G}\left(z\right).\end{array}$

The marginal df’s of the observed $X^{\prime}s$ and $Y^{\prime}s,$
respectively denoted by $F$ and $G,$ are equal to

$F\left(x\right):=p^{-1}\int_{0}^{x}\overline{\mathbf{G}}\left(z\right)d\mathbf{F}\left(z\right)\text{
and
}G\left(y\right):=p^{-1}\int_{0}^{y}\mathbf{F}\left(z\right)d\mathbf{G}\left(z\right).$

It follows that the corresponding tails are

$\overline{F}\left(x\right)=-p^{-1}\int_{x}^{\infty}\overline{\mathbf{G}}\left(z\right)d\overline{\mathbf{F}}\left(z\right)\text{
and
}\overline{G}\left(y\right)=-p^{-1}\int_{y}^{\infty}\mathbf{F}\left(z\right)d\overline{\mathbf{G}}\left(z\right).$
(1.2)

It is clear that the asymptotic behavior of $\overline{F}$ simultaneously
depends on $\overline{\mathbf{G}}$ and $\overline{\mathbf{F}},$ while that of
$\overline{G}$ only relies on $\overline{\mathbf{G}}\mathbf{.}$ Making use of
Proposition B.1.10 in de Haan and Ferreira (2006), for the regularly varying
functions $\overline{\mathbf{F}}$ and $\overline{\mathbf{G}},$ we may readily
show that both $\overline{G}$ and $\overline{F}$ are regularly varying at
infinity as well, with respective tail indices $\gamma_{2}$ and
$\gamma:=\gamma_{1}\gamma_{2}/\left(\gamma_{1}+\gamma_{2}\right).$ It is worth
noting that the issue of analyzing extreme values in the context of random
truncation, is at an early stage. Indeed, the first contribution was made in
the recent paper of Gardes and Stupfler (2015), where the authors exploited
the above relation between the three indices to define an estimator of
$\gamma_{1}$ by considering the classical Hill estimators of $\gamma$ and
$\gamma_{2}$ (Hill, 1975) as functions of two distinct numbers of top
statistics. However, they did not handle the case where these numbers are
equal because of the difficulty in assessing the dependence between the two
Hill estimators. In the present work, we introduce a tail product-limit
process for which we provide a weighted Gaussian approximation as well. This
tool will be very helpful when dealing with the estimation of any tail related
quantity. In particular, it will lead to the asymptotic normality of the
extreme value index estimator that we define, under random right-truncation,
as a function of a single sample fraction of upper order statistics. But,
prior to describing our estimation methodology, let us note that, as mentioned
by Gardes and Stupfler (2015), in order to ensure that it remains enough
extreme data for the inference to be accurate, we need to impose the condition
$\gamma_{1}<\gamma_{2}.$ In other words, we consider the situation where the
tail of the rv of interest $\mathbf{X}$ is not too contaminated by the
truncation rv $\mathbf{Y.}$ Since $\mathbf{F}$ and $\mathbf{G}$ are heavy-
tailed, then their right endpoints are infinite and thus they are equal.
Hence, from Woodroofe (1985), we may write

$\int_{x}^{\infty}\frac{d\mathbf{F}\left(y\right)}{\mathbf{F}\left(y\right)}=\int_{x}^{\infty}\frac{dF\left(y\right)}{C\left(y\right)},$
(1.3)

where

$C\left(z\right):=\mathbf{P}\left(X\leq z\leq
Y\right)=F\left(z\right)-G\left(z\right).$ (1.4)

Differentiating $\left(\ref{int}\right)$ leads to the following crucial
equation

$C\left(x\right)d\mathbf{F}\left(x\right)=\mathbf{F}\left(x\right)dF\left(x\right),$
(1.5)

(see, for instance, Strzalkowska-Kominiak and Stute, 2009), whose solution is
defined by
$\mathbf{F}\left(x\right)=\exp-\int_{x}^{\infty}dF\left(z\right)/C\left(z\right).$
Replacing $F$ and $C$ by their respective empirical counterparts

$F_{n}\left(x\right):=n^{-1}\sum\limits_{i=1}^{n}\mathbf{1}\left(X_{i}\leq
x\right)\text{ and
}C_{n}\left(x\right):=n^{-1}\sum\limits_{i=1}^{n}\mathbf{1}\left(X_{i}\leq
x\leq Y_{i}\right),$

yields the product-limit estimator

$\mathbf{F}_{n}\left(x\right):=\prod_{i:X_{i}>x}\exp\left\\{-\frac{1}{nC_{n}\left(X_{i}\right)}\right\\},$

to the underlying df $\mathbf{F.}$ The first mathematical investigation on
this estimator may be attributed to Woodroofe (1985) and the central limit
theorem under random truncation was established by Stute and Wang (2008). Note
that the approximation $\exp\left(-t\right)\sim 1-t,$ for small $t>0,$ results
in the well-known estimator introduced by Lynden-Bell (1971). Let us now
introduce a tail product-limit process corresponding to $\mathbf{F}_{n}\ $as
follows:

$\mathbf{D}_{n}\left(x\right):=\sqrt{k}\left(\frac{\overline{\mathbf{F}}_{n}\left(xX_{n-k:n}\right)}{\overline{\mathbf{F}}_{n}\left(X_{n-k:n}\right)}-x^{-1/\gamma_{1}}\right),\text{
}x>0,$ (1.6)

where $X_{1:n}\leq...\leq X_{1:n}$ denote the order statistics pertaining to
$X_{1},...,X_{n}$ and $k=k_{n}$ is a sequence of discrete rv’s satisfying

$1<k<n,\text{ }k\rightarrow\infty\text{ and }k/n\rightarrow 0\text{ as
}n\rightarrow\infty.$ (1.7)

Observe that, in the case of complete data we have $\mathbf{F}_{n}\equiv
F_{n}$ with $\overline{F}_{n}\left(X_{n-k:n}\right)=k/n$ and thus the process
defined in $\left(\ref{KM-P}\right)$ becomes

$D_{n}\left(x\right):=\sqrt{k}\left(\dfrac{n}{k}\overline{F}_{n}\left(xX_{n-k:n}\right)-x^{-1/\gamma_{1}}\right).$

By jointly applying Theorems 2.4.8 and 5.1.4 (pages 52 and 161) in de Haan and
Ferreira (2006) we have that, for $x_{0}>0$ and $0<\xi<1/2,$

$\sup_{x\geq
x_{0}}x^{(1/2-\xi)/\gamma_{1}}\left|D_{n}\left(x\right)-\Gamma\left(x;W\right)-x^{-1/\gamma_{1}}\dfrac{x^{\tau_{1}/\gamma_{1}}-1}{\tau_{1}\gamma_{1}}\sqrt{k}A_{0}\left(n/k\right)\right|\overset{\mathbf{P}}{\rightarrow}0,$
(1.8)

provided that $F$ fulfills the second-order regular variation condition with
auxiliary function $A_{0}$ tending to zero, not changing sign near infinity,
having a regularly varying absolute value with index $\tau_{1}<0$ and
satisfying $\sqrt{k}A_{0}\left(n/k\right)=O\left(1\right).$ Here
$\Gamma\left(x;W\right):=W\left(x^{-1/\gamma_{1}}\right)-x^{-1/\gamma_{1}}W\left(1\right)$
with $\left\\{W\left(s\right);\text{ }0\leq s\leq 1\right\\}$ being a standard
Wiener process. Many authors used this approximation to establish the limit
distributions of several statistics of heavy-tailed distributions, such as
tail index estimators (see, e.g., de Haan and Ferreira, 2006, page 76) and
goodness-of-fit statistics (Koning and Peng,, 2008). The main goal of this
paper is to provide an analogous result to $\left(\ref{TP-C}\right)$ in the
random truncation setting through the tail product-limit process
$\left(\ref{KM-P}\right),$ which, to the best of our knowledge, was not
addressed yet in the extreme value theory literature.

The rest of the paper is organized as follows. In Section 2, we present our
main result which consists in a Gaussian approximation to the tail product-
limit process $\mathbf{D}_{n}\left(x\right).$ As an application, we introduce,
in Section 3, a new Hill-type estimator (Hill, 1975) for the tail index
$\gamma_{1}\ $and we establish its consistency and asymptotic normality. The
finite sample behavior of the proposed estimator is checked by simulation in
Section 4. The proofs are postponed to Section 5 and some results that are
instrumental to our needs are gathered in two lemmas in the Appendix.

##  2\. Main results

Weak approximations of extreme value theory based statistics are achieved in
the second-order framework (see de Haan and Stadtmüller, 1996). Thus, it seems
quite natural to suppose that df’s $\mathbf{F}$ and $\mathbf{G}$ satisfy the
well-known second-order condition of regular variation that we express in
terms of the tail quantile functions. That is, we assume that for $x>0,$ we
have

$\underset{t\rightarrow\infty}{\lim}\dfrac{\mathbb{U}_{\mathbf{F}}\left(tx\right)/\mathbb{U}_{\mathbf{F}}\left(t\right)-x^{\gamma_{1}}}{\mathbf{A}_{\mathbf{F}}\left(t\right)}=x^{\gamma_{1}}\dfrac{x^{\tau_{1}}-1}{\tau_{1}},$
(2.9)

and

$\underset{t\rightarrow\infty}{\lim}\dfrac{\mathbb{U}_{\mathbf{G}}\left(tx\right)/\mathbb{U}_{\mathbf{G}}\left(t\right)-x^{\gamma_{2}}}{\mathbf{A}_{\mathbf{G}}\left(t\right)}=x^{\gamma_{2}}\dfrac{x^{\tau_{2}}-1}{\tau_{2}},$
(2.10)

where $\tau_{1},\tau_{2}<0$ are the second-order parameters and
$\mathbf{A}_{\mathbf{F}},$ $\mathbf{A}_{\mathbf{G}}$ are functions tending to
zero and not changing signs near infinity with regularly varying absolute
values at infinity with indices $\tau_{1},$ $\tau_{2}$ respectively. For any
df $K,$ the function
$\mathbb{U}_{K}\left(t\right):=K^{\leftarrow}\left(1-1/t\right),$ $t>1,$
stands for the tail quantile function.

######  Theorem 2.1.

Assume that both second-order conditions $(\ref{second-order})$ and
$(\ref{second-orderG})$ hold with $\gamma_{1}<\gamma_{2}.$ Let $k=k_{n}$ be a
sequence satisfying $(\ref{K}),$ then there exist a function
$\mathbf{A}_{0}\left(t\right)\sim\mathbf{A}_{\mathbf{F}}\left(1/\overline{\mathbf{F}}\left(\mathbb{U}_{F}\left(t\right)\right)\right)$
and a standard Wiener process $\left\\{\mathbf{W}\left(s\right);\text{ }0\leq
s\leq 1\right\\},$ defined on the probability space
$\left(\Omega,\mathcal{A},\mathbf{P}\right),$ such that, for
$0<\xi<1/2-\gamma/\gamma_{2}$ and $x_{0}>0,$ we have

$\sup_{x\geq
x_{0}}x^{\left(1/2-\xi\right)/\gamma-1/\gamma_{2}}\left|\mathbf{D}_{n}\left(x\right)-\mathbf{\Gamma}\left(x;\mathbf{W}\right)-x^{-1/\gamma_{1}}\dfrac{x^{\tau_{1}/\gamma_{1}}-1}{\gamma_{1}\tau_{1}}\sqrt{k}\mathbf{A}_{0}\left(n/k\right)\right|\overset{\mathbf{P}}{\rightarrow}0,$

as $n\rightarrow\infty,$ provided that
$\sqrt{k}\mathbf{A}_{0}\left(n/k\right)=O\left(1\right),$ where$\mathbb{\
}\left\\{\Gamma\left(x;\mathbf{W}\right);\text{ }x>0\right\\}$ is a Gaussian
process defined by

$\displaystyle\mathbf{\Gamma}\left(x;\mathbf{W}\right)\begin{tabular}[c]{l}$:=$\end{tabular}\frac{\gamma}{\gamma_{1}}x^{-1/\gamma_{1}}\left\\{x^{1/\gamma}\mathbf{W}\left(x^{-1/\gamma}\right)-\mathbf{W}\left(1\right)\right\\}$
$\displaystyle\ \ \ \ \ \
+\frac{\gamma}{\gamma_{1}+\gamma_{2}}x^{-1/\gamma_{1}}\int_{0}^{1}s^{-\gamma/\gamma_{2}-1}\left\\{x^{1/\gamma}\mathbf{W}\left(x^{-1/\gamma}s\right)-\mathbf{W}\left(s\right)\right\\}ds.$

######  Remark 2.1.

A very large value of $\gamma_{2}$ yields a $\gamma$-value that is very close
to $\gamma_{1},$ meaning that the really observed sample is almost the whole
dataset. In other words, the complete data case corresponds to the situation
when $1/\gamma_{2}\equiv 0,$ in which case we have $\gamma\equiv\gamma_{1}.$
It follows that

$\frac{\gamma}{\gamma_{1}+\gamma_{2}}\int_{0}^{1}s^{-\gamma/\gamma_{2}-1}\left\\{x^{1/\gamma}\mathbf{W}\left(x^{-1/\gamma}s\right)-\mathbf{W}\left(s\right)\right\\}ds\equiv
0,$

and therefore
$\Gamma\left(x;\mathbf{W}\right)=\mathbf{W}\left(x^{-1/\gamma_{1}}\right)-x^{-1/\gamma_{1}}\mathbf{W}\left(1\right),$
which agrees with the weak approximation $(\ref{TP-C}).$

##  3\. Tail index estimation

We start the construction of our estimator by noting that from Theorem 1.2.2
in de Haan and Ferreira (2006), the first-order condition $(\ref{RV-1})$ (for
$\overline{\mathbf{F}}$) implies that

$\lim_{t\rightarrow\infty}\frac{1}{\overline{\mathbf{F}}\left(t\right)}\int_{t}^{\infty}x^{-1}\overline{\mathbf{F}}\left(x\right)dx=\gamma_{1},$

which, by an integration by parts, becomes

$\lim_{t\rightarrow\infty}\frac{1}{\overline{\mathbf{F}}\left(t\right)}\int_{t}^{\infty}\log\dfrac{x}{t}d\mathbf{F}(x)=\gamma_{1}.$
(3.11)

Replacing $\mathbf{F}$ by $\mathbf{F}_{n}$ and letting $t=X_{n-k:n}$ yields

$\widehat{\gamma}_{1}:=\frac{1}{\overline{\mathbf{F}}_{n}\left(X_{n-k:n}\right)}\int_{X_{n-k:n}}^{\infty}\log\dfrac{x}{X_{n-k:n}}d\mathbf{F}_{n}\left(x\right),$

as a new estimator to $\gamma_{1}.$ By setting
$\varphi_{n}^{\left(1\right)}\left(x\right):=\mathbf{1}\left\\{x\geq
X_{n-k:n}\right\\}\log\left(x/X_{n-k:n}\right)$ and
$\varphi_{n}^{\left(2\right)}\left(x\right):=\mathbf{1}\left\\{x\geq
X_{n-k:n}\right\\},$ this may be rewritten into

$\widehat{\gamma}_{1}=\frac{\int_{0}^{\infty}\varphi_{n}^{\left(1\right)}\left(x\right)d\mathbf{F}_{n}\left(x\right)}{\int_{0}^{\infty}\varphi_{n}^{\left(2\right)}\left(x\right)d\mathbf{F}_{n}\left(x\right)}.$

From the empirical counterpart of equation $(\ref{df})$ we get

$\int_{0}^{\infty}\varphi_{n}^{\left(1\right)}\left(x\right)d\mathbf{F}_{n}\left(x\right)=\frac{1}{n}\sum_{i=n-k}^{n}\frac{\mathbf{F}_{n}\left(X_{i:n}\right)}{C_{n}\left(X_{i:n}\right)}\log\left(X_{i:n}/X_{n-k:n}\right),$

and

$\int_{0}^{\infty}\varphi_{n}^{\left(2\right)}\left(x\right)d\mathbf{F}_{n}\left(x\right)=\frac{1}{n}\sum_{i=n-k}^{n}\dfrac{\mathbf{F}_{n}\left(X_{i:n}\right)}{C_{n}\left(X_{i:n}\right)}.$

Finally, changing $i$ to $n-i+1$ yields

$\widehat{\gamma}_{1}=\left(\sum\limits_{i=1}^{k}\frac{\mathbf{F}_{n}\left(X_{n-i+1:n}\right)}{C_{n}\left(X_{n-i+1:n}\right)}\right)^{-1}\sum_{i=1}^{k}\frac{\mathbf{F}_{n}\left(X_{n-i+1:n}\right)}{C_{n}\left(X_{n-i+1:n}\right)}\log\frac{X_{n-i+1:n}}{X_{n-k:n}}.$

######  Remark 3.1.

For complete data, we have
$\mathbf{F}_{n}\mathbf{\equiv}F_{n}\mathbf{\equiv}C_{n}$ and consequently
$\widehat{\gamma}_{1}$ reduces to the classical Hill estimator (Hill, 1975).

######  Theorem 3.1.

Assume that $\left(\ref{RV-1}\right)$ holds with $\gamma_{1}<\gamma_{2}$ and
let $k=k_{n}$ be an integer sequence satisfying $(\ref{K}).$ Then
$\widehat{\gamma}_{1}\rightarrow\gamma_{1}$ in probability. Assume further
that both second-order conditions $(\ref{second-order})$ and $(\ref{second-
orderG})$ hold and
$\sqrt{k}\mathbf{A}_{\mathbf{0}}\left(n/k\right)=O\left(1\right),$ then

$\displaystyle\sqrt{k}\left(\widehat{\gamma}_{1}-\gamma_{1}\right)$
$\displaystyle=\frac{\sqrt{k}\mathbf{A}_{0}\left(n/k\right)}{1-\tau_{1}}-\gamma\mathbf{W}\left(1\right)$
$\displaystyle+\frac{\gamma}{\gamma_{1}+\gamma_{2}}\int_{0}^{1}\left(\gamma_{2}-\gamma_{1}-\gamma\log
s\right)s^{-\gamma/\gamma_{2}-1}\mathbf{W}\left(s\right)ds+o_{\mathbf{P}}\left(1\right).$

######  Corollary 3.1.

If, in addition to the assumptions of Theorem 3.1, we suppose that
$\sqrt{k}\mathbf{A}_{\mathbf{0}}\left(n/k\right)\rightarrow\lambda,$ then

$\sqrt{k}\left(\widehat{\gamma}_{1}-\gamma_{1}\right)\overset{\mathcal{D}}{\rightarrow}\mathcal{N}\left(\frac{\lambda}{1-\tau_{1}},\sigma^{2}\right),\text{
as }n\rightarrow\infty,$

where

$\sigma^{2}:=\gamma^{2}\left(1+\gamma_{1}/\gamma_{2}\right)\left(1+\left(\gamma_{1}/\gamma_{2}\right)^{2}\right)/\left(1-\gamma_{1}/\gamma_{2}\right)^{3}.$

######  Remark 3.2.

In the case of complete data we have, from Remark $\ref{remak1},$
$\sigma^{2}\equiv\gamma_{1}^{2}.$ It follows that
$\sqrt{k}\left(\widehat{\gamma}_{1}-\gamma_{1}\right)\overset{\mathcal{D}}{\rightarrow}\mathcal{N}\left(\lambda/\left(1-\tau_{1}\right),\gamma_{1}^{2}\right),$
as $n\rightarrow\infty,$ which meets the asymptotic normality of the classical
Hill estimator (Hill, 1975), see for instance, Theorem 3.2.5 in de Haan and
Ferreira (2006).

##  4\. Simulation study

This study, just intended for illustrating the performance of our estimator,
is realized through two sets of truncated and truncation data, both drawn from
Burr’s model:

$\overline{\mathbf{F}}\left(x\right)=\left(1+x^{1/\delta}\right)^{-\delta/\gamma_{1}},\text{
}\overline{\mathbf{G}}\left(x\right)=\left(1+x^{1/\delta}\right)^{-\delta/\gamma_{2}},\text{
}x\geq 0,$

where $\delta,\gamma_{1},\gamma_{2}>0.$ The corresponding percentage of
observed data is equal to $p=\gamma_{2}/(\gamma_{1}+\gamma_{2}).$ We fix
$\delta=1/4$ and choose the values $0.6$ and $0.8$ for $\gamma_{1}$ and
$70\%,$ $80\%$ and $90\%$ for $p.$ For each couple
$\left(\gamma_{1},p\right),$ we solve the equation
$p=\gamma_{2}/(\gamma_{1}+\gamma_{2})$ to get the pertaining
$\gamma_{2}$-value. We vary the common size $N$ of both samples
$\left(\mathbf{X}_{1},...,\mathbf{X}_{N}\right)$ and
$\left(\mathbf{Y}_{1},...,\mathbf{Y}_{N}\right),$ then for each size, we
generate $1000$ independent replicates. Our overall results are taken as the
empirical means of the results obtained through all repetitions. To determine
the optimal number (that we denote by $k^{\ast})$ of upper order statistics
used in the computation of $\widehat{\gamma}_{1},$ we apply the algorithm of
(Reiss and Thomas, 2007, page 137). The performance of the newly defined
estimator, in terms of absolute bias and root of the mean squared error (rmse)
is summarized in Table 4.1, where we see that, as expected, the size of the
initial sample influences the estimation: the larger $N,$ the better the
estimation. On the other hand, we note that the estimation accuracy decreases
when the truncation percentage increases, which seems logical. Finally, we
observe that the estimation of the larger value of the tail index is less
precise.

${\small p=0.7}$ $\gamma_{1}=0.6$ $\gamma_{1}=0.8$ $N$ $n$ $k^{\ast}$ absolute
bias rmse $n$ $k^{\ast}$ absolute bias rmse ${\small 200}$ $139$ $8$ $0.1811$
$0.4645$ $140$ $9$ $0.2485$ $0.6034$ ${\small 300}$ $210$ $17$ $0.1280$
$0.3451$ $209$ $23$ $0.1230$ $0.5082$ ${\small 500}$ $348$ $28$ $0.1151$
$0.2803$ $348$ $31$ $0.1024$ $0.3708$ ${\small 1000}$ $699$ $43$ $0.0461$
$0.2421$ $700$ $42$ $0.0684$ $0.3825$ $1500$ $1050$ $67$ $0.0212$ $0.2362$
$1050$ $57$ $0.0527$ $0.2539$ $2000$ $1399$ $95$ $0.0254$ $0.2261$ $1401$ $94$
$0.0469$ $0.2602$ ${\small p=0.8}$ ${\small 200}$ $159$ $10$ $0.0759$ $0.5235$
$159$ $11$ $0.1435$ $0.6815$ ${\small 300}$ $240$ $21$ $0.0485$ $0.3647$ $238$
$18$ $0.1052$ $0.4511$ ${\small 500}$ $399$ $34$ $0.0403$ $0.2718$ $400$ $41$
$0.0935$ $0.3170$ ${\small 1000}$ $800$ $55$ $0.0408$ $0.1999$ $800$ $54$
$0.0504$ $0.2881$ ${\small 1500}$ $1205$ $116$ $0.0562$ $0.1911$ $1199$ $85$
$0.0428$ $0.2362$ ${\small 2000}$ $1599$ $117$ $0.0285$ $0.1534$ $1599$ $117$
$0.0373$ $0.2054$ ${\small p=0.9}$ ${\small 200}$ $180$ $14$ $0.0537$ $0.5204$
$179$ $17$ $0.1098$ $0.7531$ ${\small 300}$ $269$ $24$ $0.0388$ $0.3343$ $268$
$26$ $0.0844$ $0.3654$ ${\small 500}$ $450$ $48$ $0.0294$ $0.2869$ $450$ $49$
$0.0721$ $0.2448$ ${\small 1000}$ $899$ $64$ $0.0359$ $0.1557$ $899$ $67$
$0.0465$ $0.2014$ ${\small 1500}$ $1349$ $103$ $0.0171$ $0.1350$ $1349$ $101$
$0.0385$ $0.1828$ ${\small 2000}$ $1799$ $144$ $0.0188$ $0.1107$ $1799$ $145$
$0.0251$ $0.1466$

Table 4.1. Absolute bias and rmse of the tail index estimator based on 1000
right-truncated samples of Burr models.

##  5\. Proofs

###  5.1. Proof of Theorem 2.1

Set $U_{i}:=\overline{F}\left(X_{i}\right)$ and define the corresponding
uniform tail empirical process by
$\alpha_{n}\left(s\right):=\sqrt{k}\left(\mathbf{U}_{n}\left(s\right)-s\right),$
for $0\leq s\leq 1,$ where
$\mathbf{U}_{n}\left(s\right):=k^{-1}\sum_{i=1}^{n}\mathbf{1}\left(U_{i}<ks/n\right).$
The weighted weak approximation to $\alpha_{n}\left(s\right)$ given in terms
of, either a sequence of Wiener processes (see, e.g., Einmahl, 1992 and Dress
et al., 2006) or a single Wiener process as in Proposition 3.1 of Einmahl et
al. (2006), will be very crucial to our proof procedure. In the sequel, we use
the latter representation which says that: there exists a Wiener process
$\mathbf{W},$ such that for every $0\leq\eta<1/2,$

$\sup_{0<s\leq
1}s^{-\eta}\left|\alpha_{n}\left(s\right)-\mathbf{W}\left(s\right)\right|\overset{\mathbf{p}}{\rightarrow}0,\text{
as }n\rightarrow\infty.$ (5.12)

We begin by fixing $x_{0}>0,$ then we decompose
$k^{-1/2}\mathbf{D}_{n}\left(x\right),$ for $x\geq x_{0},$ as the sum of the
following four terms:

$\mathbf{M}_{n1}\left(x\right):=x^{-1/\gamma_{1}}\frac{\overline{\mathbf{F}}_{n}\left(xX_{n-k:n}\right)-\overline{\mathbf{F}}\left(xX_{n-k:n}\right)}{\overline{\mathbf{F}}\left(xX_{n-k:n}\right)},\vskip
6.0pt plus 2.0pt minus 2.0pt$
$\mathbf{M}_{n2}\left(x\right):=-\frac{\overline{\mathbf{F}}\left(xX_{n-k:n}\right)}{\overline{\mathbf{F}}_{n}\left(X_{n-k:n}\right)}\frac{\overline{\mathbf{F}}_{n}\left(X_{n-k:n}\right)-\overline{\mathbf{F}}\left(X_{n-k:n}\right)}{\overline{\mathbf{F}}\left(X_{n-k:n}\right)},\vskip
6.0pt plus 2.0pt minus 2.0pt$
$\mathbf{M}_{n3}\left(x\right):=\left(\frac{\overline{\mathbf{F}}\left(xX_{n-k:n}\right)}{\overline{\mathbf{F}}_{n}\left(X_{n-k:n}\right)}-x^{-1/\gamma_{1}}\right)\frac{\overline{\mathbf{F}}_{n}\left(xX_{n-k:n}\right)-\overline{\mathbf{F}}\left(xX_{n-k:n}\right)}{\overline{\mathbf{F}}\left(xX_{n-k:n}\right)}\vskip
6.0pt plus 2.0pt minus 2.0pt$

and

$\mathbf{M}_{n4}\left(x\right):=\frac{\overline{\mathbf{F}}\left(xX_{n-k:n}\right)}{\overline{\mathbf{F}}\left(X_{n-k:n}\right)}-x^{-1/\gamma_{1}}.$

In order to establish the result of the theorem, we will successively show
that, under the first-order of regular variation conditions, we have uniformly
on $x\geq x_{0},$ for $\gamma/\gamma_{2}<\eta<1/2$ and $\epsilon>0$
sufficiently small

$\displaystyle x^{1/\gamma_{1}}\sqrt{k}\mathbf{M}_{n1}\left(x\right)$
$\displaystyle=x^{1/\gamma}\left\\{\frac{\gamma}{\gamma_{1}}\mathbf{W}\left(x^{-1/\gamma}\right)+\frac{\gamma}{\gamma_{1}+\gamma_{2}}\int_{0}^{1}t^{-\gamma/\gamma_{2}-1}\mathbf{W}\left(x^{-1/\gamma}t\right)dt\right\\}+O_{\mathbf{p}}\left(\epsilon\right)x^{\left(1-\eta\right)/\gamma\pm\epsilon},\vskip
6.0pt plus 2.0pt minus 2.0pt$
$x^{1/\gamma_{1}}\sqrt{k}\mathbf{M}_{n2}\left(x\right)=-\left\\{\frac{\gamma}{\gamma_{1}}\mathbf{W}\left(1\right)+\frac{\gamma}{\gamma_{1}+\gamma_{2}}\int_{0}^{1}t^{-\gamma/\gamma_{2}-1}\mathbf{W}\left(t\right)dt\right\\}+O_{\mathbf{p}}\left(\epsilon\right)x^{\pm\epsilon},\vskip
6.0pt plus 2.0pt minus 2.0pt$

and

$x^{1/\gamma_{1}}\sqrt{k}\mathbf{M}_{n3}\left(x\right)=O_{\mathbf{p}}\left(\epsilon\right)x^{-1/\gamma_{1}+\left(1-\eta\right)/\gamma\pm\epsilon}.$

Moreover, if we assume the second-order condition we will show that

$x^{1/\gamma_{1}}\sqrt{k}\mathbf{M}_{n4}\left(x\right)=\left(1+o_{\mathbf{p}}\left(1\right)\right)\dfrac{x^{\tau_{1}/\gamma_{1}}-1}{\gamma_{1}\tau_{1}}\sqrt{k}\mathbf{A}_{0}\left(n/k\right).$

Here $O_{\mathbf{p}}$ and $o_{\mathbf{p}}$ stand for the usual stochastic
order symbols. For convenience, let $a_{k}:=\mathbb{U}_{F}\left(n/k\right)$
and recall that $\mathbb{U}_{F}$ is regularly varying (with index $\gamma).$
Then by combining Corollary 2.2.2 with Proposition B.1.10 in de Haan and
Ferreira (2006), we show that
$X_{n-k:n}/a_{k}\overset{\mathbf{P}}{\rightarrow}1$ as $n\rightarrow\infty,$
which implies, due to the regular variation of $\overline{\mathbf{F}},$ that
$\overline{\mathbf{F}}\left(xa_{k}\right)/\overline{\mathbf{F}}\left(xX_{n-k:n}\right)$
$=1+o_{\mathbf{p}}\left(1\right)$ and therefore

$\mathbf{M}_{n1}\left(x\right)=\left(1+o_{\mathbf{p}}\left(1\right)\right)\mathbf{M}_{n1}^{\ast}\left(x\right),$
(5.13)

where

$\mathbf{M}_{n1}^{\ast}\left(x\right):=x^{-1/\gamma_{1}}\frac{\overline{\mathbf{F}}_{n}\left(xX_{n-k:n}\right)-\overline{\mathbf{F}}\left(xX_{n-k:n}\right)}{\overline{\mathbf{F}}\left(xa_{k}\right)}.$

Now, observe that, in view of equation $(\ref{df}),$ we may write

$\mathbf{F}\left(x\right)=\exp-\Lambda\left(x\right)\text{ and
}\mathbf{F}_{n}\left(x\right)=\exp-\Lambda_{n}\left(x\right),$

where $\Lambda\left(x\right)$ and its empirical counterpart
$\Lambda_{n}\left(x\right)$ are defined by
$\int_{x}^{\infty}dF\left(z\right)/C\left(z\right)$ and
$\int_{x}^{\infty}dF_{n}\left(z\right)/C_{n}\left(z\right)$ respectively. Note
that $\overline{\mathbf{F}}_{n}\left(xX_{n-k:n}\right),$
$\overline{\mathbf{F}}\left(xX_{n-k:n}\right)$ and
$\overline{\mathbf{F}}\left(xa_{k}\right)$ tend to zero in probability,
uniformly on $x\geq x_{0},$ it follows that
$\Lambda_{n}\left(xX_{n-k:n}\right),$ $\Lambda\left(xX_{n-k:n}\right)$ and
$\Lambda\left(xa_{k}\right)$ go to zero in probability as well. Using the
approximation $1-\exp(-t)\sim t,$ as $t\downarrow 0,$ we may write

$x^{1/\gamma_{1}}\mathbf{M}_{n1}^{\ast}\left(x\right)=\left(1+o_{\mathbf{p}}\left(1\right)\right)\frac{\Lambda_{n}\left(xX_{n-k:n}\right)-\Lambda\left(xX_{n-k:n}\right)}{\Lambda\left(xa_{k}\right)}.$

Next, we provide a Gaussian approximation to the expression

$\sqrt{k}\dfrac{\Lambda_{n}\left(xX_{n-k:n}\right)-\Lambda\left(xX_{n-k:n}\right)}{\Lambda\left(xa_{k}\right)},$

then we deduce one to
$\sqrt{k}x^{1/\gamma_{1}}\mathbf{M}_{n1}^{\ast}\left(x\right).$ For this, we
decompose the difference
$\Lambda_{n}\left(xX_{n-k:n}\right)-\Lambda\left(xX_{n-k:n}\right)$ in the sum
of

$S_{n1}\left(x\right):=-\int_{xa_{k}}^{\infty}\frac{d\left(\overline{F}_{n}\left(z\right)-\overline{F}\left(z\right)\right)}{C\left(z\right)},$
$S_{n2}\left(x\right):=-\int_{xX_{n-k:n}}^{\infty}\left\\{\frac{1}{C_{n}\left(z\right)}-\frac{1}{C\left(z\right)}\right\\}d\overline{F}_{n}\left(z\right),$

and

$S_{n3}\left(x\right):=-\int_{xX_{n-k:n}}^{xa_{k}}\frac{d\left(\overline{F}_{n}\left(z\right)-\overline{F}\left(z\right)\right)}{C\left(z\right)}.$

For the first term, we use the fact that $\overline{F}_{n}\left(z\right)=0$
for $z\geq X_{n:n},$ to write, after an integration by parts and a change of
variables,
$S_{n1}\left(x\right)=S_{n1}^{\left(1\right)}\left(x\right)-S_{n1}^{\left(2\right)}\left(x\right),$
with

$S_{n1}^{\left(1\right)}\left(x\right):=\frac{\overline{F}_{n}\left(a_{k}x\right)-\overline{F}\left(a_{k}x\right)}{C\left(a_{k}x\right)}\text{
and
}S_{n1}^{\left(2\right)}\left(x\right):=\int_{x}^{\infty}\frac{\overline{F}_{n}\left(a_{k}z\right)-\overline{F}\left(a_{k}z\right)}{C^{2}\left(a_{k}z\right)}dC\left(a_{k}z\right).$

It is easy to verify that
$\overline{F}_{n}\left(xa_{k}\right)-\overline{F}\left(xa_{k}\right)=\dfrac{\sqrt{k}}{n}\alpha_{n}\left(\dfrac{n}{k}\overline{F}\left(xa_{k}\right)\right),$
it follows that

$\frac{\sqrt{k}S_{n1}^{\left(1\right)}\left(x\right)}{\Lambda\left(a_{k}x\right)}=d_{n}\left(x\right)\alpha_{n}\left(\frac{n}{k}\overline{F}\left(a_{k}x\right)\right),$

where
$d_{n}\left(x\right):=\dfrac{k/n}{\Lambda\left(xa_{k}\right)C\left(a_{k}x\right)}.$
From Lemma 6.2 $\left(iii\right),$ we have

$d_{n}\left(x\right)=\left(\gamma/\gamma_{1}\right)x^{1/\gamma}+O\left(\epsilon\right)x^{1/\gamma\pm\epsilon},$
(5.14)

as $n\rightarrow\infty,$ uniformly on $x\geq x_{0},$ it follows that

$\frac{\sqrt{k}S_{n1}^{\left(1\right)}\left(x\right)}{\Lambda\left(a_{k}x\right)}=\left\\{\left(\gamma/\gamma_{1}\right)x^{1/\gamma}+O_{\mathbf{p}}\left(\epsilon\right)x^{1/\gamma\pm\epsilon}\right\\}\alpha_{n}\left(\frac{n}{k}\overline{F}\left(a_{k}x\right)\right).$

On the other hand, for $0<\eta<1/2,$ the sequence of rv’s $\sup_{0<s\leq
1}\left|\alpha_{n}\left(s\right)\right|/s^{\eta}$ is stochastically bounded.
This comes from the inequality

$\sup_{0<s\leq
1}s^{-\eta}\left|\alpha_{n}\left(s\right)\right|\leq\sup_{0<s\leq
1}s^{-\eta}\left|\alpha_{n}\left(s\right)-\mathbf{W}\left(s\right)\right|+\sup_{0<s\leq
1}s^{-\eta}\left|\mathbf{W}\left(s\right)\right|,$

with approximation $\left(\ref{approx}\right)$ and the fact $\sup_{0<s\leq
1}s^{-\eta}\left|\mathbf{W}\left(s\right)\right|=O_{\mathbf{p}}\left(1\right)$
(see, e.g., Lemma 3.2 in Einmahl et al., 2006). Now, let $\epsilon>0$ be
sufficiently small. Then, by applying Potter’s inequalities to $\overline{F}$
(see, e.g., Proposition B.1.9, assertion 5 in de Haan and Ferreira, 2006), we
write
$\dfrac{n}{k}\overline{F}\left(a_{k}x\right)\leq\left(1+\epsilon\right)x^{-1/\gamma\pm\epsilon},$
it follows that
$\alpha_{n}\left(\dfrac{n}{k}\overline{F}\left(a_{k}x\right)\right)=O_{\mathbf{p}}\left(x^{-\eta/\gamma\pm\eta\epsilon}\right).$
For notational simplicity and without loss of generality, we attribute
$\epsilon$ to any constant times $\epsilon$ and $v^{\pm\epsilon}$ to any
linear combinations of $v^{\pm c_{1}\epsilon}$ and $v^{\pm c_{2}\epsilon},\
$for every $c_{1},c_{2}>0.$ Therefore

$\frac{\sqrt{k}S_{n1}^{\left(1\right)}\left(x\right)}{\Lambda\left(a_{k}x\right)}=\frac{\gamma}{\gamma_{1}}x^{1/\gamma}\alpha_{n}\left(\frac{n}{k}\overline{F}\left(a_{k}x\right)\right)+O_{\mathbf{p}}\left(\epsilon\right)x^{\left(1-\eta\right)/\gamma\pm\epsilon}.$

For $S_{n1}^{\left(2\right)}\left(x\right),$ let us write

$\frac{\sqrt{k}S_{n1}^{\left(2\right)}\left(x\right)}{\Lambda\left(a_{k}x\right)}=d_{n}\left(x\right)\frac{C\left(a_{k}x\right)}{C\left(a_{k}\right)}\int_{x}^{\infty}\dfrac{C^{2}\left(a_{k}\right)}{C^{2}\left(a_{k}z\right)}\alpha_{n}\left(\dfrac{n}{k}\overline{F}\left(a_{k}z\right)\right)d\dfrac{C\left(a_{k}z\right)}{C\left(a_{k}\right)}.$

From Lemma 6.2 $\left(i\right),$ the function $C$ is regularly varying at
infinity with index $\left(-1/\gamma_{2}\right),$ as $\overline{G}$ is, this
implies that
$C\left(xa_{k}\right)/C\left(a_{k}\right)=x^{-1/\gamma_{2}}+O\left(\epsilon\right)x^{-1/\gamma_{2}\pm\epsilon}.$
Then by using $\left(\ref{dn}\right),$ we get

$d_{n}\left(x\right)\frac{C\left(a_{k}x\right)}{C\left(a_{k}\right)}=\left(\gamma/\gamma_{1}\right)x^{1/\gamma_{1}}+O\left(\epsilon\right)x^{1/\gamma_{1}\pm\epsilon},\text{
as }n\rightarrow\infty.$ (5.15)

For convenience, we set
$\sqrt{k}S_{n1}^{\left(2\right)}\left(x\right)/\Lambda\left(a_{k}x\right)=\left\\{1+O\left(\epsilon\right)x^{\pm\epsilon}\right\\}\mathcal{T}_{n}\left(x\right),$
where

$\mathcal{T}_{n}\left(x\right):=\frac{\gamma}{\gamma_{1}}x^{1/\gamma_{1}}\int_{x}^{\infty}\dfrac{C^{2}\left(a_{k}\right)}{C^{2}\left(a_{k}z\right)}\alpha_{n}\left(\dfrac{n}{k}\overline{F}\left(a_{k}z\right)\right)d\dfrac{C\left(a_{k}z\right)}{C\left(a_{k}\right)},$

which we decompose in the sum of

$I_{n}\left(x\right):=\frac{\gamma}{\gamma_{1}}x^{1/\gamma_{1}}\int_{x}^{\infty}\dfrac{C^{2}\left(a_{k}\right)}{C^{2}\left(a_{k}z\right)}\alpha_{n}\left(\dfrac{n}{k}\overline{F}\left(a_{k}z\right)\right)d\dfrac{F\left(a_{k}z\right)}{C\left(a_{k}\right)},$
$J_{n}\left(x\right):=-\frac{\gamma}{\gamma_{1}}x^{1/\gamma_{1}}\int_{x}^{\infty}\left\\{\dfrac{C^{2}\left(a_{k}\right)}{C^{2}\left(a_{k}z\right)}-z^{2/\gamma_{2}}\right\\}\alpha_{n}\left(\dfrac{n}{k}\overline{F}\left(a_{k}z\right)\right)d\dfrac{G\left(a_{k}z\right)}{C\left(a_{k}\right)},$

and

$K_{n}\left(x\right):=\frac{\gamma}{\gamma_{1}}x^{1/\gamma_{1}}\int_{x}^{\infty}z^{2/\gamma_{2}}\alpha_{n}\left(\dfrac{n}{k}\overline{F}\left(a_{k}z\right)\right)d\dfrac{\overline{G}\left(a_{k}z\right)}{C\left(a_{k}\right)}.$

Recall that $a_{k}\rightarrow\infty,$
$C\left(a_{k}\right)\sim\overline{G}\left(a_{k}\right)$ and
$\overline{F}\left(a_{k}\right)=o\left(\overline{G}\left(a_{k}\right)\right)$
as $n\rightarrow\infty.$ On the other hand, by using, once again, Potter’s
inequalities to $C,$ (regularly varying at infinity with index
$-1/\gamma_{2}),$ we write, for all large $n$ and $z\geq x,$

$\left(1-\epsilon\right)z^{-1/\gamma_{2}}\min\left(z^{\epsilon},z^{-\epsilon}\right)\leq\dfrac{C\left(a_{k}z\right)}{C\left(a_{k}\right)}\leq\left(1+\epsilon\right)z^{-1/\gamma_{2}}\max\left(z^{\epsilon},z^{-\epsilon}\right).$
(5.16)

It is clear this implies that
$C^{2}\left(a_{k}\right)/C^{2}\left(a_{k}z\right)\leq\left(1-\epsilon\right)^{-2}z^{2/\gamma_{2}\pm
2\epsilon}.$ In view of the stochastic boundedness of $\sup_{0<s\leq
1}\left|\alpha_{n}\left(s\right)\right|/s^{\eta}$ and the fact that
$\dfrac{n}{k}\overline{F}\left(a_{k}z\right)\leq\left(1+\epsilon\right)z^{-1/\gamma\pm\epsilon},$
we have

$I_{n}\left(x\right)=o_{\mathbf{p}}\left(1\right)x^{1/\gamma_{1}}\int_{x}^{\infty}z^{2/\gamma_{2}\mp
2\epsilon}\left(z^{-1/\gamma\pm\epsilon}\right)^{\eta}d\dfrac{\overline{F}\left(a_{k}z\right)}{\overline{F}\left(a_{k}\right)}.$

Integrating by parts, we readily get
$I_{n}\left(x\right)=o_{\mathbf{p}}\left(1\right)x^{1/\gamma_{2}-\eta/\gamma\pm\epsilon}=o_{\mathbf{p}}\left(1\right)x^{\left(1-\eta\right)/\gamma\pm\epsilon}.$
Let us now consider $J_{n}\left(x\right).$ From Proposition B.1.10 in de Haan
and Ferreira (2006), we have
$\left|C\left(a_{k}z\right)/C\left(a_{k}\right)-z^{-1/\gamma_{2}}\right|\leq\epsilon
z^{-1/\gamma_{2}\pm\epsilon}.$ Applying the mean value theorem, then combining
this inequality with $\left(\ref{C_inequa1}\right),$ yield

$\left|\dfrac{C^{2}\left(a_{k}\right)}{C^{2}\left(a_{k}z\right)}-z^{2/\gamma_{2}}\right|\leq\epsilon\frac{2\left(z^{\pm\epsilon}+1\right)}{\left(1-\epsilon\right)^{3}}z^{2/\gamma_{2}\pm\epsilon}.$

Similar arguments as the above lead to
$J_{n}\left(x\right)=O_{\mathbf{p}}\left(\epsilon\right)x^{\left(1-\eta\right)/\gamma\pm\epsilon}.$
Now, we focus on $K_{n}\left(x\right).$ Since
$C\left(a_{k}\right)\sim\overline{G}\left(a_{k}\right),$ then

$K_{n}\left(x\right)=\left(1+o_{\mathbf{p}}\left(1\right)\right)\frac{\gamma}{\gamma_{1}}x^{1/\gamma_{1}}\int_{x}^{\infty}z^{2/\gamma_{2}}\alpha_{n}\left(\dfrac{n}{k}\overline{F}\left(a_{k}z\right)\right)d\dfrac{\overline{G}\left(a_{k}z\right)}{\overline{G}\left(a_{k}\right)}.$

Let $G^{\leftarrow}$ denote the quantile function pertaining to df $G$ and use
the change of variables
$z=G^{\leftarrow}\left(1-s\overline{G}\left(a_{k}\right)\right)/a_{k}$ to get

$K_{n}\left(x\right)=-\left(1+o_{\mathbf{p}}\left(1\right)\right)\frac{\gamma}{\gamma_{1}}x^{1/\gamma_{1}}\int_{0}^{\tfrac{\overline{G}\left(a_{k}x\right)}{\overline{G}\left(a_{k}\right)}}\left(\frac{G^{\leftarrow}\left(1-s\overline{G}\left(a_{k}\right)\right)}{a_{k}}\right)^{2/\gamma_{2}}\alpha_{n}\left(\ell_{n}\left(s\right)\right)ds,$

where
$\ell_{n}\left(s\right):=\dfrac{n}{k}\overline{F}\left(G^{\leftarrow}\left(1-s\overline{G}\left(a_{k}\right)\right)\right).$
It is easy to check that

$K_{n}\left(x\right)=-\left(1+o_{\mathbf{p}}\left(1\right)\right)\sum_{i=1}^{3}K_{ni}\left(x\right),$

where

$K_{n1}\left(x\right):=\frac{\gamma}{\gamma_{1}}x^{\frac{1}{\gamma_{1}}}\int_{0}^{\tfrac{\overline{G}\left(a_{k}x\right)}{\overline{G}\left(a_{k}\right)}}\left\\{\left(\frac{G^{\leftarrow}\left(1-s\overline{G}\left(a_{k}\right)\right)}{a_{k}}\right)^{2/\gamma_{2}}-s^{-2}\right\\}\alpha_{n}\left(\ell_{n}\left(s\right)\right)ds,$
$K_{n2}\left(x\right):=\frac{\gamma}{\gamma_{1}}x^{1/\gamma_{1}}\int_{x^{-1/\gamma_{2}}}^{\tfrac{\overline{G}\left(a_{k}x\right)}{\overline{G}\left(a_{k}\right)}}s^{-2}\alpha_{n}\left(\ell_{n}\left(s\right)\right)ds,$

and

$K_{n3}\left(x\right):=\frac{\gamma}{\gamma_{1}}x^{1/\gamma_{1}}\int_{0}^{x^{-1/\gamma_{2}}}s^{-2}\alpha_{n}\left(\ell_{n}\left(s\right)\right)ds.$

By routine manipulations and similar arguments based on stochastic boundedness
of $\sup_{0<s\leq 1}\left|\alpha_{n}\left(s\right)\right|/s^{\eta}$ and the
aforementioned Proposition B.1.10 applied to the regularly varying functions
$\overline{G}$ and $G^{\leftarrow}\left(1-\cdot\right),$ we show that
$K_{ni}\left(x\right)=O_{\mathbf{p}}\left(\epsilon\right)x^{\left(1-\eta\right)/\gamma\mp\epsilon},$
$i=1,2$ and
$K_{n3}\left(x\right)=O_{\mathbf{p}}\left(1\right)x^{\left(1-\eta\right)/\gamma\mp\epsilon},$
therefore we omit the details. Up to this stage, we have shown that
$\mathcal{T}_{n}\left(x\right)=O_{\mathbf{p}}\left(1\right)x^{\left(1-\eta\right)/\gamma\mp\epsilon}.$
It follows that

$\frac{\sqrt{k}S_{n1}^{\left(2\right)}\left(x\right)}{\Lambda\left(a_{k}x\right)}=\mathcal{T}_{n}\left(x\right)+O_{\mathbf{p}}\left(\epsilon\right)x^{\left(1-\eta\right)/\gamma\mp\epsilon},$

which, after gathering the components of $\mathcal{T}_{n}\left(x\right),$ is
equal to

$\frac{\gamma}{\gamma_{1}}x^{1/\gamma_{1}}\int_{0}^{x^{-1/\gamma_{2}}}s^{-2}\alpha_{n}\left(\ell_{n}\left(s\right)\right)ds+O_{\mathbf{p}}\left(\epsilon\right)x^{\left(1-\eta\right)/\gamma\mp\epsilon}.$

Therefore

$\displaystyle\frac{\sqrt{k}S_{n1}\left(x\right)}{\Lambda\left(a_{k}x\right)}$
$\displaystyle=\frac{\gamma}{\gamma_{1}}x^{1/\gamma}\alpha_{n}\left(\frac{n}{k}\overline{F}\left(a_{k}x\right)\right)$
$\displaystyle-\frac{\gamma}{\gamma_{1}}x^{1/\gamma_{1}}\int_{0}^{x^{-1/\gamma_{2}}}s^{-2}\alpha_{n}\left(\ell_{n}\left(s\right)\right)ds+O_{\mathbf{p}}\left(\epsilon\right)x^{\left(1-\eta\right)/\gamma\mp\epsilon}.$

Recall that $\gamma_{1}<\gamma_{2}$ and
$\gamma/\gamma_{2}=\gamma_{1}/\left(\gamma_{1}+\gamma_{2}\right),$ then we may
choose the constant $\eta$ in such a way that $\gamma/\gamma_{2}<\eta<1/2.$
Making use of weak approximation $(\ref{approx}),$ we obtain

$\displaystyle\frac{\sqrt{k}S_{n1}\left(x\right)}{\Lambda\left(a_{k}x\right)}$
$\displaystyle=\frac{\gamma}{\gamma_{1}}x^{1/\gamma}\mathbf{W}\left(\frac{n}{k}\overline{F}\left(a_{k}x\right)\right)+\frac{\gamma}{\gamma_{1}}x^{1/\gamma}{\displaystyle\int_{0}^{x^{-1/\gamma_{2}}}}s^{-2}\mathbf{W}\left(\ell_{n}\left(s\right)\right)ds+O_{\mathbf{p}}\left(\epsilon\right)x^{\left(1-\eta\right)/\gamma\mp\epsilon}.$

Note that
$k/n=\overline{F}\left(G^{\leftarrow}\left(1-\overline{G}\left(a_{k}\right)\right)\right),$
hence

$\ell_{n}\left(s\right)=\frac{\overline{F}\left(G^{\leftarrow}\left(1-s\overline{G}\left(a_{k}\right)\right)\right)}{\overline{F}\left(G^{\leftarrow}\left(1-\overline{G}\left(a_{k}\right)\right)\right)}.$

Since $s\rightarrow\overline{F}\circ G^{\leftarrow}\left(1-s\right)$ is
regularly varying at infinity with index $\gamma_{2}/\gamma,$ then, from
Proposition B.1.10 in de Haan and Ferreira (2006), we have for all large $n$

$\omega_{n}\left(s\right):=\left|\ell_{n}\left(s\right)-s^{\gamma_{2}/\gamma}\right|\leq\epsilon
s^{\gamma_{2}/\gamma\pm\epsilon}.$ (5.17)

Recall that $x_{0}>0$ is fixed, then $\sup_{x\geq x_{0}}\sup_{0<s\leq
x^{-1/\gamma_{2}}}\omega_{n}\left(s\right)\rightarrow 0,$ as
$n\rightarrow\infty.$ On the other hand, by using Levy’s modulus of continuity
of the Wiener process (see, e.g., Theorem 1.1.1 in Csörgő and Révész,, 1981),
we have

$\left|\mathbf{W}\left(\ell_{n}\left(s\right)\right)-\mathbf{W}\left(s^{\gamma_{2}/\gamma}\right)\right|\leq
2\sqrt{\omega_{n}\left(s\right)\log\left(1/\omega_{n}\left(s\right)\right)},$

uniformly on $s\geq x^{-1/\gamma_{2}},$ almost surely. By using the fact that,
$\log u<\epsilon u^{-\epsilon}$ as $u\downarrow 0,$ together with inequality
$(\ref{ln}),$ we get
$\left|\mathbf{W}\left(\ell_{n}\left(s\right)\right)-\mathbf{W}\left(s^{\gamma_{2}/\gamma}\right)\right|\leq
2\epsilon s^{\left(\gamma_{2}/\gamma\right)\left(1-\epsilon\right)/2}.$
Following our convention, we may write that
$\left(\gamma_{2}/\gamma\pm\epsilon\right)\left(1-\epsilon/2\right)\equiv\gamma_{2}/\gamma\pm\epsilon.$
Since $\gamma_{1}<\gamma_{2}$ then $\gamma_{2}/\left(2\gamma\right)>1$ and
after elementary calculation, we show that uniformly on $x\geq x_{0}$

$\frac{\gamma}{\gamma_{1}}x^{1/\gamma_{1}}\int_{0}^{x^{-1/\gamma_{2}}}s^{-2}\mathbf{W}\left(\ell_{n}\left(s\right)\right)ds=\frac{\gamma}{\gamma_{1}}x^{1/\gamma_{1}}\int_{0}^{x^{-1/\gamma_{2}}}s^{-2}\mathbf{W}\left(s^{\gamma_{2}/\gamma}\right)ds+O_{\mathbf{p}}\left(\epsilon\right)x^{1/\left(2\gamma\right)\pm\epsilon}.$

By similar arguments, we get

$\frac{\gamma}{\gamma_{1}}x^{1/\gamma}\mathbf{W}\left(\frac{n}{k}\overline{F}\left(a_{k}x\right)\right)=\frac{\gamma}{\gamma_{1}}x^{1/\gamma}\mathbf{W}\left(x^{-1/\gamma}\right)+O_{\mathbf{p}}\left(\epsilon\right)x^{1/\left(2\gamma\right)\pm\epsilon}.$

It is obvious that
$O_{\mathbf{p}}\left(\epsilon\right)x^{1/\left(2\gamma\right)\pm\epsilon}+O_{\mathbf{p}}\left(\epsilon\right)x^{\left(1-\eta\right)/\gamma\pm\epsilon}=O_{\mathbf{p}}\left(\epsilon\right)x^{\left(1-\eta\right)/\gamma\pm\epsilon},$
it follows that

$\frac{\sqrt{k}S_{n1}\left(x\right)}{\Lambda\left(a_{k}x\right)}=\frac{\gamma}{\gamma_{1}}x^{1/\gamma}\mathbf{W}\left(x^{-1/\gamma}\right)+\frac{\gamma}{\gamma_{1}}x^{1/\gamma_{1}}\int_{0}^{x^{-1/\gamma_{2}}}s^{-2}\mathbf{W}\left(s^{\gamma_{2}/\gamma}\right)ds+O_{\mathbf{p}}\left(\epsilon\right)x^{\left(1-\eta\right)/\gamma\pm\epsilon}.$

After a change of variables, this may be rewritten into

$\displaystyle\frac{\sqrt{k}S_{n1}\left(x\right)}{\Lambda\left(a_{k}x\right)}$
(5.18)
$\displaystyle=\frac{\gamma}{\gamma_{1}}x^{1/\gamma}\mathbf{W}\left(x^{-1/\gamma}\right)+\frac{\gamma}{\gamma_{1}+\gamma_{2}}x^{1/\gamma}\int_{0}^{1}t^{-\gamma/\gamma_{2}-1}\mathbf{W}\left(x^{-1/\gamma}t\right)dt+O_{\mathbf{p}}\left(\epsilon\right)x^{\left(1-\eta\right)/\gamma\pm\epsilon}.$

Now, we consider the second term $S_{n2}\left(x\right).\ $We have
$\overline{F}_{n}\left(z\right)=0,$ for $z\geq X_{n:n},$ thus

$S_{n2}\left(x\right)=\int_{xX_{n-k:n}}^{X_{n:n}}\frac{C_{n}\left(z\right)-C\left(z\right)}{C_{n}\left(z\right)C\left(z\right)}d\overline{F}_{n}\left(z\right).$

Therefore

$\left|S_{n2}\left(x\right)\right|\leq\theta_{n}\int_{xX_{n-k:n}}^{\infty}\frac{\left|C_{n}\left(z\right)-C\left(z\right)\right|}{C^{2}\left(z\right)}dF_{n}\left(z\right),$

where $\theta_{n}:=\sup_{X_{1:n}\leq z\leq
X_{n:n}}\left\\{C\left(z\right)/C_{n}\left(z\right)\right\\},$ which is
stochastically bounded (see, e.g., Stute and Wang, 2008). By recalling that
$C=\overline{G}-\overline{F}$ and $C_{n}=\overline{G}_{n}-\overline{F}_{n},$
with $G_{n}$ denoting the empirical df of $G,$ we write
$\left|S_{n2}\left(x\right)\right|\leq\theta_{n}\left(T_{n1}\left(x\right)+T_{n2}\left(x\right)\right),$
where

$T_{n1}\left(x\right):=\int_{xX_{n-k:n}}^{\infty}\dfrac{\left|\overline{F}_{n}\left(z\right)-\overline{F}\left(z\right)\right|}{C^{2}\left(z\right)}dF_{n}\left(z\right)$

and

$T_{n2}\left(x\right):=\int_{xX_{n-k:n}}^{\infty}\dfrac{\left|\overline{G}_{n}\left(z\right)-\overline{G}\left(z\right)\right|}{C^{2}\left(z\right)}dF_{n}\left(z\right).$

It is easy to verify that, by a change of variables, we have

$\displaystyle\frac{\sqrt{k}T_{n1}\left(x\right)}{\Lambda\left(a_{k}x\right)}$
$\displaystyle=d_{n}\left(x\right)\frac{k/n}{C\left(a_{k}\right)}\frac{C\left(a_{k}x\right)}{C\left(a_{k}\right)}$
$\displaystyle\times\frac{C^{2}\left(a_{k}\right)}{C^{2}\left(xX_{n-k:n}\right)}\int_{1}^{\infty}\dfrac{\left|\alpha_{n}\left(\dfrac{n}{k}\overline{F}\left(xX_{n-k:n}z\right)\right)\right|}{C^{2}\left(xX_{n-k:n}z\right)/C^{2}\left(xX_{n-k:n}\right)}d\frac{F_{n}\left(xX_{n-k:n}z\right)}{\overline{F}\left(a_{k}\right)}.$

Recall that, uniformly on $x\geq x_{0},$ we have
$C\left(a_{k}\right)/C\left(xX_{n-k:n}\right)=O_{\mathbf{p}}\left(1\right)x^{1/\gamma_{2}\pm\epsilon}.$
Moreover, we use $\left(\ref{C_inequa1}\right)$ and $\left(\ref{dn2}\right)$
to write

$\displaystyle\frac{\sqrt{k}T_{n1}\left(x\right)}{\Lambda\left(a_{k}x\right)}$
$\displaystyle=O_{\mathbf{p}}\left(\frac{k/n}{C\left(a_{k}\right)}\right)x^{1/\gamma\pm\epsilon}$
$\displaystyle\times\int_{1}^{\infty}z^{2/\gamma_{2}}\left|\alpha_{n}\left(\dfrac{n}{k}\overline{F}\left(xX_{n-k:n}z\right)\right)\right|d\frac{F_{n}\left(xX_{n-k:n}z\right)}{\overline{F}\left(a_{k}\right)}.$

On the other hand, by using the stochastic boundedness of $\sup_{0<s\leq
1}\left|\alpha_{n}\left(s\right)\right|/s^{\eta}$ we get

$\frac{\sqrt{k}T_{n1}\left(x\right)}{\Lambda\left(a_{k}x\right)}=O_{\mathbf{p}}\left(\frac{k/n}{C\left(a_{k}\right)}\right)x^{\left(1-\eta\right)/\gamma\pm\epsilon}\int_{1}^{\infty}z^{2/\gamma_{2}-\eta/\gamma\pm\epsilon}d\frac{\overline{F}_{n}\left(xX_{n-k:n}z\right)}{\overline{F}\left(a_{k}\right)},$

where the integral may be split as follows

$\int_{1}^{\infty}z^{2/\gamma_{2}-\eta/\gamma\pm\epsilon}d\frac{\overline{F}_{n}\left(xX_{n-k:n}z\right)}{\overline{F}\left(a_{k}\right)}=P_{n}\left(x\right)+Q_{n}\left(x\right),$

where

$P_{n}\left(x\right):=\int_{1}^{\infty}z^{2/\gamma_{2}-\eta/\gamma\pm\epsilon}d\left\\{\frac{\overline{F}_{n}\left(xX_{n-k:n}z\right)-\overline{F}\left(xX_{n-k:n}z\right)}{\overline{F}\left(a_{k}\right)}\right\\},$

and

$Q_{n}\left(x\right):=\int_{1}^{\infty}z^{2/\gamma_{2}-\eta/\gamma\pm\epsilon}d\frac{\overline{F}\left(xX_{n-k:n}z\right)}{\overline{F}\left(a_{k}\right)}.$

It is clear that

$P_{n}\left(x\right)=k^{-1/2}\int_{1}^{\infty}z^{2/\gamma_{2}-\eta/\gamma\pm\epsilon}d\alpha_{n}\left(\frac{n}{k}\overline{F}\left(xX_{n-k:n}z\right)\right).$

By similar arguments as those used above, we show that
$P_{n}\left(x\right)=o_{\mathbf{p}}\left(x^{-\eta/\gamma\pm\epsilon}\right)$
and $Q_{n}\left(x\right)=O_{\mathbf{p}}\left(x^{-1/\gamma\pm\epsilon}\right).$
Therefore

$\frac{\sqrt{k}T_{n1}\left(x\right)}{\Lambda\left(a_{k}x\right)}=x^{-\eta/\gamma\pm\epsilon}O_{\mathbf{p}}\left(\frac{k/n}{C\left(a_{k}\right)}\right).$

Next, let $V_{i}:=\overline{G}\left(Y_{i}\right),$ $i=1,...,n,$ and define the
corresponding tail empirical process
$\beta_{n}\left(s\right):=\sqrt{k}\left(\mathbf{V}_{n}\left(s\right)-s\right),$
for $0\leq s\leq 1,$ where
$\mathbf{V}_{n}\left(s\right):=k^{-1}\sum_{i=1}^{n}\mathbf{1}\left(V_{i}<ks/n\right).$
Like for $\alpha_{n}\left(s\right),$ we also have $\sup_{0<s\leq
1}\left|\beta_{n}\left(s\right)\right|/s^{\eta}=O_{\mathbf{p}}\left(1\right),$
therefore by similar arguments as those used for $T_{n1}\left(x\right),$ with
the facts that $\overline{G}\left(t\right)\sim C\left(t\right)$ as
$t\rightarrow\infty$ and $\gamma_{2}>\gamma,$ we show that

$\frac{\sqrt{k}T_{n2}\left(x\right)}{\Lambda\left(a_{k}x\right)}=O_{\mathbf{p}}\left(\frac{k/n}{C^{1-\eta}\left(a_{k}\right)}\right)x^{\left(1-\eta\right)/\gamma\pm\epsilon}.$

From Lemma 6.2 $\left(ii\right)$, we have that both
$\dfrac{n}{k}C\left(a_{k}\right)$ and
$\dfrac{n}{k}C^{1-\eta}\left(a_{k}\right)$ tend to infinity, it follows that

$\frac{\sqrt{k}T_{n1}\left(x\right)}{\Lambda\left(a_{k}x\right)}=o_{\mathbf{p}}\left(x^{-\eta/\gamma\pm\epsilon}\right)\text{
and
}\frac{\sqrt{k}T_{n2}\left(x\right)}{\Lambda\left(a_{k}x\right)}=o_{\mathbf{p}}\left(x^{\left(1-\eta\right)/\gamma\pm\epsilon}\right).$

Since
$o_{\mathbf{p}}\left(x^{-\eta/\gamma\pm\epsilon}\right)+o_{\mathbf{p}}\left(x^{\left(1-\eta\right)/\gamma\pm\epsilon}\right)=o_{\mathbf{p}}\left(x^{\left(1-\eta\right)/\gamma\pm\epsilon}\right),$
then

$\frac{\sqrt{k}S_{n2}\left(x\right)}{\Lambda\left(a_{k}x\right)}=o_{\mathbf{p}}\left(x^{\left(1-\eta\right)/\gamma\pm\epsilon}\right).$
(5.19)

Let us now focus on the third term $S_{n3},$ which, by integration by parts,
equals the sum of

$S_{n3}^{\left(1\right)}\left(x\right):=-\int_{xX_{n-k:n}}^{xa_{k}}\frac{\overline{F}_{n}\left(z\right)-\overline{F}\left(z\right)}{C^{2}\left(z\right)}dC\left(z\right),$

and

$S_{n3}^{\left(2\right)}\left(x\right)=-\frac{\overline{F}_{n}\left(a_{k}x\right)-\overline{F}\left(a_{k}x\right)}{C\left(a_{k}x\right)}+\frac{\overline{F}_{n}\left(xX_{n-k:n}\right)-\overline{F}\left(xX_{n-k:n}\right)}{C\left(xX_{n-k:n}\right)}.$

By using the change of variables $z=txa_{k}$ we get

$\frac{\sqrt{k}S_{n3}^{\left(1\right)}\left(x\right)}{\Lambda\left(a_{k}x\right)}=-d_{n}\left(x\right)\int_{X_{n-k:n}/a_{k}}^{1}\frac{\alpha_{n}\left(\dfrac{n}{k}\overline{F}\left(a_{k}xz\right)\right)}{\left(C\left(a_{k}xz\right)/C\left(a_{k}x\right)\right)^{2}}d\frac{C\left(a_{k}xz\right)}{C\left(a_{k}x\right)},$

and

$\frac{\sqrt{k}S_{n3}^{\left(2\right)}\left(x\right)}{\Lambda\left(a_{k}x\right)}=-d_{n}\left(x\right)\left\\{\alpha_{n}\left(\frac{n}{k}\overline{F}\left(a_{k}x\right)\right)-\frac{C\left(a_{k}x\right)}{C\left(xX_{n-k:n}\right)}\alpha_{n}\left(\dfrac{n}{k}\overline{F}\left(xX_{n-k:n}\right)\right)\right\\}.$

Routine manipulations, including Proposition B.1.10 in de Haan and Ferreira
(2006) and the stochastic boundedness of $\sup_{0<s\leq
1}\left|\alpha_{n}\left(s\right)\right|/s^{\eta},$ yield

$\frac{\sqrt{k}S_{n3}^{\left(1\right)}\left(x\right)}{\Lambda\left(a_{k}x\right)}=o_{\mathbf{p}}\left(x^{\left(1-\eta\right)/\gamma\pm\epsilon}\right)\text{
and
}\frac{\sqrt{k}S_{n3}^{\left(2\right)}\left(x\right)}{\Lambda\left(a_{k}x\right)}=o_{\mathbf{p}}\left(x^{\left(1-\eta\right)/\gamma\pm\epsilon}\right).$

It follows that

$\sqrt{k}S_{n3}\left(x\right)/\Lambda\left(a_{k}x\right)=o_{\mathbf{p}}\left(x^{\left(1-\eta\right)/\gamma\pm\epsilon}\right).$
(5.20)

By gathering results $\left(\ref{Sn1}\right),$ $\left(\ref{Sn2}\right)$ and
$\left(\ref{Sn3}\right),$ we obtain

$\displaystyle\sqrt{k}\frac{\Lambda_{n}\left(xX_{n-k:n}\right)-\Lambda\left(xX_{n-k:n}\right)}{\Lambda\left(a_{k}x\right)}$
(5.21)
$\displaystyle=\frac{\gamma}{\gamma_{1}}x^{1/\gamma}\mathbf{W}\left(x^{-1/\gamma}\right)+\frac{\gamma}{\gamma_{1}+\gamma_{2}}x^{1/\gamma}\int_{0}^{1}t^{-\gamma/\gamma_{2}-1}\mathbf{W}\left(x^{-1/\gamma}t\right)dt+O_{\mathbf{p}}\left(\epsilon\right)x^{\left(1-\eta\right)/\gamma\pm\epsilon},$

which yields that

$\displaystyle x^{1/\gamma_{1}}\sqrt{k}\mathbf{M}_{n1}^{\ast}\left(x\right)$
$\displaystyle=x^{1/\gamma}\left\\{\frac{\gamma}{\gamma_{1}}\mathbf{W}\left(x^{-1/\gamma}\right)+\frac{\gamma}{\gamma_{1}+\gamma_{2}}\int_{0}^{1}t^{-\gamma/\gamma_{2}-1}\mathbf{W}\left(x^{-1/\gamma}t\right)dt\right\\}+O_{\mathbf{p}}\left(\epsilon\right)x^{\left(1-\eta\right)/\gamma\pm\epsilon}.$

We show that the expectation of the absolute value of the first term in the
right-hand side of the previous equation equals
$O_{\mathbf{p}}\left(x^{1/\left(2\gamma\right)}\right).$ Since
$1/\left(2\gamma\right)<\left(1-\eta\right)/\gamma,$ we have
$x^{1/\gamma_{1}}\sqrt{k}\mathbf{M}_{n1}^{\ast}\left(x\right)=O_{\mathbf{p}}\left(x^{\left(1-\eta\right)/\gamma\pm\epsilon}\right),$
which leads to

$x^{1/\gamma_{1}}\sqrt{k}\mathbf{M}_{n1}\left(x\right)=x^{1/\gamma_{1}}\sqrt{k}\mathbf{M}_{n1}^{\ast}\left(x\right)+o_{\mathbf{p}}\left(x^{\left(1-\eta\right)/\gamma\pm\epsilon}\right).$

Recall that $\epsilon>0$ is chosen sufficiently small, then for any
$0<\eta<1/2,$ we have

$\displaystyle x^{1/\gamma_{1}}\sqrt{k}\mathbf{M}_{n1}\left(x\right)$
$\displaystyle=x^{1/\gamma}\left\\{\frac{\gamma}{\gamma_{1}}\mathbf{W}\left(x^{-1/\gamma}\right)+\frac{\gamma}{\gamma_{1}+\gamma_{2}}\int_{0}^{1}t^{-\gamma/\gamma_{2}-1}\mathbf{W}\left(x^{-1/\gamma}t\right)dt\right\\}+O_{\mathbf{p}}\left(\epsilon\right)x^{\left(1-\eta\right)/\gamma\pm\epsilon}.$

Before we treat the term $\mathbf{M}_{n2}\left(x\right),$ it is worth
mentioning that by letting $x=1$ in the previous approximation, we infer that

$\frac{\overline{\mathbf{F}}_{n}\left(X_{n-k:n}\right)}{\overline{\mathbf{F}}\left(X_{n-k:n}\right)}-1=O_{\mathbf{p}}\left(k^{-1/2}\right)=o_{\mathbf{p}}\left(1\right),\text{
}\left(k\rightarrow\infty\right).$ (5.22)

This, with the regular variation of $\overline{\mathbf{F}},$ imply that

$\frac{\overline{\mathbf{F}}\left(xX_{n-k:n}\right)}{\overline{\mathbf{F}}_{n}\left(X_{n-k:n}\right)}=\left(1+O_{\mathbf{p}}\left(x^{\pm\epsilon}\right)\right)x^{-1/\gamma_{1}}.$
(5.23)

To represent $\sqrt{k}\mathbf{M}_{n2}\left(x\right),$ we apply results
$\left(\ref{Lambda}\right)$ (for $x=1)$ and $\left(\ref{important2}\right)$ to
get

$x^{1/\gamma_{1}}\sqrt{k}\mathbf{M}_{n2}\left(x\right)=-\left\\{\frac{\gamma}{\gamma_{1}}\mathbf{W}\left(1\right)+\frac{\gamma}{\gamma_{1}+\gamma_{2}}\int_{0}^{1}t^{-\gamma/\gamma_{2}-1}\mathbf{W}\left(t\right)dt\right\\}+O_{\mathbf{p}}\left(\epsilon\right)x^{\pm\epsilon}.$

For the third term $\mathbf{M}_{n3}\left(x\right),$ we write

$x^{1/\gamma_{1}}\sqrt{k}\mathbf{M}_{n3}\left(x\right)=\left(\frac{\overline{\mathbf{F}}\left(xX_{n-k:n}\right)}{\overline{\mathbf{F}}_{n}\left(X_{n-k:n}\right)}-x^{-1/\gamma_{1}}\right)x^{1/\gamma_{1}}\sqrt{k}\mathbf{M}_{n1}\left(x\right),$

which, by equation $\left(\ref{important2}\right),$ is equal to
$O_{\mathbf{p}}\left(\epsilon\right)x^{-1/\gamma_{1}+\left(1-\eta\right)/\gamma\pm\epsilon}.$
Let $\eta_{0}$ be such that $\gamma/\gamma_{2}<\eta_{0}<\eta<1/2,$ then
$\eta_{0}-\eta<0$ and for $\epsilon>0$ sufficiently small, we have
$\left(\eta_{0}-\eta\right)/\gamma+\epsilon<0.$ Since $x\geq x_{0}>0,$ then
$O_{\mathbf{p}}\left(\epsilon\right)x^{\left(\eta_{0}-\eta\right)/\gamma\pm\epsilon}=O_{\mathbf{p}}\left(\epsilon\right)$
and thus

$x^{1/\gamma_{1}-\left(1-\eta_{0}\right)/\gamma}\left\\{\sqrt{k}\left(\mathbf{M}_{n1}\left(x\right)+\mathbf{M}_{n2}\left(x\right)+\mathbf{M}_{n3}\left(x\right)\right)-\mathbf{\Gamma}\left(x;\mathbf{W}\right)\right\\}=O_{\mathbf{p}}\left(\epsilon\right),$
(5.24)

where $\mathbf{\Gamma}\left(x;\mathbf{W}\right)$ is the Gaussian process given
in Theorem 2.1. For the fourth term $\mathbf{M}_{n4}\left(x\right),$ it
suffices to use the uniform inequality to second-order condition
$\left(\ref{second-order}\right),$ given in assertion (2.3.23) of Theorem
2.3.9 in de Haan and Ferreira (2006), to get

$\sqrt{k}\mathbf{M}_{n4}\left(x\right)=\left(1+o_{\mathbf{p}}\left(1\right)\right)x^{-1/\gamma_{1}}\dfrac{x^{\tau_{1}/\gamma_{1}}-1}{\gamma_{1}\tau_{1}}\sqrt{k}\widetilde{\mathbf{A}}_{\mathbf{F}}\left(1/\overline{\mathbf{F}}\left(X_{n-k:n}\right)\right),$

for a possibly different function $\widetilde{\mathbf{A}}_{\mathbf{F}}$ with
$\widetilde{\mathbf{A}}_{\mathbf{F}}\sim\mathbf{A}_{\mathbf{F}}.$ Then
Proposition B.1.10 in de Haan and Ferreira (2006) and the fact that
$t\rightarrow\widetilde{\mathbf{A}}_{\mathbf{F}}\left(1/\overline{\mathbf{F}}\left(\mathbb{U}_{F}\left(t\right)\right)\right)=:\mathbf{A}_{0}$
$\left(t\right)$ is regularly varying with index $\tau_{1}/\gamma_{1}$ with
$X_{n-k:n}/a_{k}\overset{\mathbf{p}}{\rightarrow}1,$ imply that

$\frac{\mathbf{A}_{0}\left(1/\overline{F}\left(X_{n-k:n}\right)\right)}{\mathbf{A}_{0}\left(1/\overline{F}\left(a_{k}\right)\right)}\overset{\mathbf{p}}{\rightarrow}1,\text{
as }n\rightarrow\infty,$

as well. Since
$o_{\mathbf{p}}\left(1\right)x^{-1/\gamma_{1}}\dfrac{x^{\tau_{1}/\gamma_{1}}-1}{\gamma_{1}\tau_{1}}=o_{\mathbf{p}}\left(x^{-1/\gamma_{1}+\left(1-\eta\right)/\gamma\pm\epsilon}\right),$
and by assumption
$\sqrt{k}\mathbf{A}_{0}\left(1/\overline{F}\left(a_{k}\right)\right)=\sqrt{k}\mathbf{A}_{0}\left(n/k\right)=O\left(1\right),$
it follows that

$\sqrt{k}\mathbf{M}_{n4}\left(x\right)=x^{-1/\gamma_{1}}\dfrac{x^{\tau_{1}/\gamma_{1}}-1}{\gamma_{1}\tau_{1}}\sqrt{k}\mathbf{A}_{0}\left(n/k\right)+o_{\mathbf{p}}\left(x^{-1/\gamma_{1}+\left(1-\eta\right)/\gamma\pm\epsilon}\right).$

Finally, by letting $\epsilon\downarrow 0$ in $\left(\ref{MM}\right),$ we end
up with

$\sup_{x\geq
x_{0}}x^{1/\gamma_{1}-\left(1-\eta_{0}\right)/\gamma}\left|\mathbf{D}_{n}\left(x\right)-\mathbf{\Gamma}\left(x;\mathbf{W}\right)-x^{-1/\gamma_{1}}\dfrac{x^{\tau_{1}/\gamma_{1}}-1}{\gamma_{1}\tau_{1}}\sqrt{k}\mathbf{A}_{0}\left(n/k\right)\right|\overset{\mathbf{P}}{\rightarrow}0,$

for every $x_{0}>0$ and $\gamma/\gamma_{2}<\eta_{0}<\eta<1/2.$ Letting
$\eta_{0}:=1/2-\xi$ and recalling that $1/\gamma_{1}=1/\gamma-1/\gamma_{2}$
yields that $0<\xi<1/2-\gamma/\gamma_{2}$ and achieves the
proof.$\hfill\square$

###  5.2. Proof of Theorem 3.1

We start by proving the consistency of $\widehat{\gamma}_{1}$ that we write as
$\widehat{\gamma}_{1}=\int_{1}^{\infty}x^{-1}\overline{\mathbf{F}}_{n}\left(xX_{n-k:n}\right)/\overline{\mathbf{F}}_{n}\left(X_{n-k:n}\right)dx.$
It is readily checked that this may be decomposed into the sum of

$I_{1n}:=\int_{1}^{\infty}x^{-1}\frac{\overline{\mathbf{F}}\left(xX_{n-k:n}\right)}{\overline{\mathbf{F}}\left(X_{n-k:n}\right)}dx\text{
and
}I_{2n}:=\int_{1}^{\infty}x^{-1}\sum_{i=1}^{3}\mathbf{M}_{ni}\left(x\right)dx.$

By the regular variation of $\overline{\mathbf{F}}$ $\left(\ref{RV-1}\right)$
and Potter’s inequalities, we get
$I_{1n}\overset{\mathbf{P}}{\rightarrow}\gamma_{1}$ as $n\rightarrow\infty.$
Then, we just need to show that $I_{2n}$ tends to zero in probability. From
$\left(\ref{MM}\right)$ we have

$I_{2n}=\frac{1}{\sqrt{k}}\int_{1}^{\infty}x^{-1}\mathbf{\Gamma}\left(x;\mathbf{W}\right)dx+\frac{1}{\sqrt{k}}\int_{1}^{\infty}x^{-1}o_{\mathbf{p}}\left(x^{\left(1-\eta\right)/\gamma-1/\gamma_{1}}\right)dx.$

On the one hand, since $\gamma/\gamma_{2}<\eta,$ the second integral above is
finite and therefore the second term of $I_{2n}$ is negligible in probability.
On the other hand, we have

$\displaystyle\int_{1}^{\infty}x^{-1}\mathbf{\Gamma}\left(x;\mathbf{W}\right)dx$
$\displaystyle=\frac{\gamma}{\gamma_{1}}\int_{1}^{\infty}x^{1/\gamma_{2}-1}\left\\{\mathbf{W}\left(x^{-1/\gamma}\right)-x^{-1/\gamma}\mathbf{W}\left(1\right)\right\\}dx+\frac{\gamma}{\gamma_{1}+\gamma_{2}}$
$\displaystyle\times\int_{1}^{\infty}x^{1/\gamma_{2}-1}\left\\{\int_{0}^{1}s^{-\gamma/\gamma_{2}-1}\left\\{\mathbf{W}\left(x^{-1/\gamma}s\right)-x^{-1/\gamma}\mathbf{W}\left(s\right)\right\\}ds\right\\}dx,$

which, after some elementary but tedious manipulations of integral calculus
(change of variables and integration by parts), becomes

$\displaystyle\int_{1}^{\infty}x^{-1}\mathbf{\Gamma}\left(x;\mathbf{W}\right)dx$
$\displaystyle=-\gamma\mathbf{W}\left(1\right)$ (5.25)
$\displaystyle+\frac{\gamma}{\gamma_{1}+\gamma_{2}}\int_{0}^{1}\left(\gamma_{2}-\gamma_{1}-\gamma\log
s\right)s^{-\gamma/\gamma_{2}-1}\mathbf{W}\left(s\right)ds.$

By using the facts that $\mathbf{E}\left|\mathbf{W}\left(s\right)\right|\leq
s^{1/2}$ and $\gamma_{1}<\gamma_{2},$ we deduce that
$\int_{1}^{\infty}x^{-1}\mathbf{\Gamma}\left(x;\mathbf{W}\right)dx$ is
stochastically bounded and therefore the first term of $I_{2n}$ is is
negligible in probability as well. Consequently, we have
$I_{2n}=o_{\mathbf{P}}\left(1\right)$ when $n\rightarrow\infty,$ as sought. As
for the Gaussian representation result, it is easy to verify that
$\sqrt{k}\left(\widehat{\gamma}_{1}-\gamma_{1}\right)=\int_{1}^{\infty}x^{-1}\mathbf{D}_{n}\left(x\right)dx.$
Then, applying Theorem 2.1 yields that

$\sqrt{k}\left(\widehat{\gamma}_{1}-\gamma_{1}\right)=\frac{\sqrt{k}\mathbf{A}_{0}\left(n/k\right)}{1-\tau}+\int_{1}^{\infty}x^{-1}\mathbf{\Gamma}\left(x;\mathbf{W}\right)dx+o_{\mathbf{P}}\left(1\right),$

and finally, using result $\left(\ref{GAMMA}\right)$ completes the
proof.$\hfill\square$

###  5.3. Proof of Corollary 3.1

We set

$\sqrt{k}\left(\widehat{\gamma}_{1}-\gamma_{1}\right)=\gamma\Delta+\frac{\sqrt{k}\mathbf{A}_{0}\left(n/k\right)}{1-\tau}+o_{\mathbf{P}}\left(1\right),$

where $\Delta:=a\Delta_{1}+b\Delta_{2}-\Delta_{3},$ with
$a:=\left(\gamma_{2}-\gamma_{1}\right)/\left(\gamma_{1}+\gamma_{2}\right),$
$b:=-\gamma/\left(\gamma_{1}+\gamma_{2}\right)$ and

$\Delta_{1}:=\int_{0}^{1}s^{\rho-2}\mathbf{W}\left(s\right)ds,\text{
}\Delta_{2}:=\int_{0}^{1}s^{\rho-2}\mathbf{W}\left(s\right)\log sds,\text{
}\Delta_{3}:=\mathbf{W}\left(1\right),$

with $\rho:=1-\gamma/\gamma_{2}>0.$

It is clear that the asymptotic mean is equal to
$\lim_{n\rightarrow\infty}\sqrt{k}\mathbf{A}_{0}\left(n/k\right)/\left(1-\tau\right),$
while for the asymptotic variance we find, after elementary but tedious
computations, the following covariances:

$\displaystyle\mathbf{E}\left[\Delta_{1}^{2}\right]$
$\displaystyle=\frac{2}{\rho\left(2\rho-1\right)},\text{
}\mathbf{E}\left[\Delta_{2}^{2}\right]=\frac{2\left(4\rho-1\right)}{\rho^{2}\left(2\rho-1\right)^{3}},\text{
}\mathbf{E}\left[\Delta_{3}^{2}\right]=1,$
$\displaystyle\mathbf{E}\left[\Delta_{1}\Delta_{2}\right]$
$\displaystyle=\frac{1-4\rho}{\rho^{2}\left(2\rho-1\right)^{2}},\text{
}\mathbf{E}\left[\Delta_{1}\Delta_{3}\right]=\frac{1}{\rho},\text{
}\mathbf{E}\left[\Delta_{2}\Delta_{3}\right]=-\frac{1}{\rho^{2}}.$

It follows that

$\mathbf{E}\left[\Delta^{2}\right]=\frac{2a^{2}}{\rho\left(2\rho-1\right)}+\frac{2b^{2}\left(4\rho-1\right)}{\rho^{2}\left(2\rho-1\right)^{3}}+\frac{2ab\left(1-4\rho\right)}{\rho^{2}\left(2\rho-1\right)^{2}}+\frac{2b}{\rho^{2}}-\frac{2a}{\rho}+1.$

Replacing $a,$ $b$ and $\rho$ by their values achieves the
proof.$\hfill\square\vskip 6.0pt plus 2.0pt minus 2.0pt$

Concluding notes

We would like to emphasize the fact that, unlike Gardes and Stupfler (2015)
who defined their estimator in terms of two (not necessarily equal) sample
fractions $k=k^{\prime}$ of upper order statistics from $X$ and $Y$
respectively, we build our estimator on the basis of just a single sample
fraction. The consideration of two distinct sample fractions poses a problem
from a computational point of view, as the issue of selecting an optimal
couple of sample fractions is not as easy and usual as determining just one
best number of top statistics to be used in parameter estimate computation.
Besides that, Gardes and Stupfler (2015) didn’t treat the asymptotic normality
when $k=k^{\prime}$ and only carried out their simulations in this very
particular case, as they mentioned in their conclusion. For these reasons, we
don’t compare the two estimators in Section 5.

A more thorough simulation study, with confidence interval construction and
eventual comparison with the estimator of Gardes and Stupfler (2015), will be
part of a future work. Another point, beyond the scope of the present paper,
that deserves to be considered is to reduce estimation biases under random
truncation. Similar anterior works were done with complete datasets by, for
instance, Peng and Qi (2004), Li and Peng (2010) and Brahimi et al. (2013).

We finish this work by making a comment on relation $\left(\ref{log}\right),$
which actually is a special case of a more general functional of the
distribution tail defined by

$\Gamma_{t}\left(g,\alpha\right):=\frac{\dfrac{1}{\overline{F}\left(t\right)}\int_{t}^{\infty}g\left(\dfrac{\overline{F}\left(x\right)}{\overline{F}\left(t-\right)}\right)\left(\log\dfrac{x}{t}\right)^{\alpha}dF\left(x\right)}{\int_{0}^{1}g\left(x\right)\left(-\log
x\right)^{\alpha}dx},\text{ }t\geq 0,$

where $g$ is some weight function and $\alpha$ some positive real number. As a
consequence of the fact that
$\lim_{t\rightarrow\infty}\Gamma_{t}\left(g,\alpha\right)=\gamma^{\alpha},$
this functional can be considered as the starting point to constructing a
whole class of estimators for distribution tail parameters. Indeed, in the
complete data case, we replace $F$ by its empirical counterpart $F_{n}$ and
$t$ by $X_{n-k:n}$ to get the following statistic which generalizes several
extreme value theory based procedures of estimation already existing in the
literature:

$\Gamma_{n,k}\left(g,\alpha\right):=\frac{\dfrac{1}{k}\sum\limits_{i=1}^{k}g\left(\dfrac{i}{k+1}\right)\left(\log\dfrac{X_{n-i+1:n}}{X_{n-k:n}}\right)^{\alpha}}{\int_{0}^{1}g\left(x\right)\left(-\log
x\right)^{\alpha}dx}.$

When $g=\alpha=1,$ we recover the famous Hill estimator (Hill, 1975). For a
detailed list of extreme value index estimators drawn from the statistic
above, we refer to the paper of Ciuperca and Mercadier (2010), where the
authors propose an estimation approach of the second-order parameter by
considering differences and quotients of several forms of
$\Gamma_{n,k}\left(g,\alpha\right).$ By analogy, when we deal with randomly
truncated observations, we substitute the product-limit estimator
$\mathbf{F}_{n}$ for $F$ in the formula of $\Gamma_{t}\left(g,\alpha\right)$
in order to obtain the following family of parameter estimators under random
truncation:

$\Gamma_{n,k}\left(g,\alpha\right):=\frac{\sum\limits_{i=1}^{k}a_{n}^{\left(i\right)}g\left(\dfrac{\overline{\mathbf{F}}_{n}\left(X_{n-i+1:n}\right)}{\overline{\mathbf{F}}_{n}\left(X_{n-k-1:n}\right)}\right)\left(\log\dfrac{X_{n-i+1:n}}{X_{n-k:n}}\right)^{\alpha}}{\sum\limits_{i=1}^{k}a_{n}^{\left(i\right)}\int_{0}^{1}g\left(x\right)\left(-\log
x\right)^{\alpha}dx}.$

This would have fruitful consequences on the statistical analysis of extremes
under random truncation.

## References

  * Brahimi et al. (2013) Brahimi, B., Meraghni, D., Necir, A., Yahia, D., 2013. A bias-reduced estimator for the mean of a heavy-tailed distribution with an infinite second moment. J. Statist. Plann. Inference 143, no. 6, 1064-1081.
  * Ciuperca and Mercadier (2010) Ciuperca, G., Mercadier, C., 2010. Semi-parametric estimation for heavy tailed distributions. Extremes 13, 55-87.
  * Csörgő and Révész, (1981) Csörgő, M., Révész, P., 1981. Strong approximations in probability and statistics. Probability and Mathematical Statistics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London.
  * Dress et al. (2006) Dree, H., de Haan, L., Li, D., 2006. Approximations to the tail empirical distribution function with application to testing extreme value conditions. J. Statist. Plann. Inference 136, 3498-3538.
  * Einmahl (1992) Einmahl, J.H.J., 1992. Limit theorems for tail processes with application to intermediate quantile estimation. J. Statist. Plann. Inference 32, 137-145.
  * Einmahl et al. (2006) Einmahl, J.H.J., de Haan, L., Li, D., 2006\. Weighted approximations of tail copula processes with application to testing the bivariate extreme value condition. Ann. Statist. 34, 1987-2014.
  * Embrechts et al. (1997) Embrechts, P., Klüppelberg, C., Mikosch, T., 1997. Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
  * Escudero and Ortega (2008) Escudero, F., Ortega, E., 2008. Actuarial comparisons for aggregate claims with randomly right-truncated claims. Insurance Math. Econom. 43, 255-262.
  * Gardes and Stupfler (2015) Gardes, L., Stupfler, G., 2015. Estimating extreme quantiles under random truncation. TEST 24, 207-227.
  * de Haan and Stadtmüller (1996) de Haan, L., Stadtmüller, U., 1996. Generalized regular variation of second order. J. Australian Math. Soc. (Series A) 61, 381-395.
  * de Haan and Ferreira (2006) de Haan, L., Ferreira, A., 2006. Extreme Value Theory: An Introduction. Springer.
  * Herbst (1999) Herbst, T., 1999. An application of randomly truncated data models in reserving IBNR claims. Insurance Math. Econom. 25, 123-131.
  * Hill (1975) Hill, B.M., 1975. A simple general approach to inference about the tail of a distribution. Ann. Statist. 3, 1163-1174.
  * Hua and Joe (2011) Hua, L., Harry, J., 2011. Second order regular variation and conditional tail expectation of multiple risks. Insurance Math. Econom. 49: 537-546.
  * Koning and Peng, (2008) Koning, A.J., Peng, L., 2008. Goodness-of-fit tests for a heavy tailed distribution. J. Statist. Plann. Inference 138, no. 12, 3960-3981.
  * Lawless (2002) Lawless, J.F., 2002. Statistical Models and Methods for Lifetime Data, Second Edition. Wiley Series in Probability and Statistics.
  * Li and Peng (2010) Li, D., Peng, L., Yang, J., 2010. Bias reduction for high quantiles. J. Statist. Plann. Inference 140, no. 9, 2433-2441.
  * Lynden-Bell (1971) Lynden-Bell, D., 1971. A method of allowing for known observational selection in small samples applied to 3CR quasars. Monthly Notices Roy. Astronom. Soc. 155: 95-118.
  * Mason (1982) Mason, D.M., 1982. Laws of large numbers for sums of extreme values. Ann. Probab. 10, 756-764.
  * Peng and Qi (2004) Peng, L., Qi, Y., 2004. Estimating the first- and second-order parameters of a heavy-tailed distribution. Aust. N. Z. J. Stat. 46, no. 2, 305-312.
  * Reiss and Thomas (2007) Reiss, R.D., Thomas, M., 2007. Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields, 3rd ed. Birkhäuser Verlag, Basel, Boston, Berlin.
  * Resnick (2006) Resnick, S., 2006. Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer.
  * Strzalkowska-Kominiak and Stute (2009) Strzalkowska-Kominiak, E., Stute, W., 2009. Martingale representations of the Lynden-Bell estimator with applications. Statist. Probab. Lett. 79: 814-820.
  * Stute and Wang (2008) Stute, W., Wang, J., 2008. The central limit theorem under random truncation. Bernoulli 14, 604-622.
  * Wang, Jewell and Tsai (1986) Wang M.C., Jewell N.P., Tsai, W.Y., 1986. Asymptotic properties of the product-limit estimate under random truncation. Ann Statist 14:1597-1605.
  * Wang (1989) Wang, M.C., 1989. A semiparametric model for randomly truncated data. Statist. Assoc., 84, 742-748
  * Woodroofe (1985) Woodroofe, M., 1985. Estimating a distribution function with truncated data. Ann. Statist. 13:163-177.

##  6\. Appendix

######  Lemma 6.1.

Assume that both second-order conditions $(\ref{second-order})$ and
$(\ref{second-orderG})$ hold. Then, for all large $x,$ there exist constants
$d_{1},d_{2}>0,$ such that

$\overline{F}\left(x\right)=\left(1+o\left(1\right)\right)d_{1}x^{-1/\gamma}\text{
and
}\overline{G}\left(x\right)=\left(1+o\left(1\right)\right)d_{2}x^{-1/\gamma_{2}}.$

###### Proof.

We only show the first statement since the second one follows by similar
arguments. To this end, we rewrite the first equation of
$\left(\ref{FbarGbar}\right)$ into

$\overline{F}\left(x\right)=-p^{-1}\overline{\mathbf{G}}\left(x\right)\overline{\mathbf{F}}\left(x\right)\int_{1}^{\infty}\frac{\overline{\mathbf{G}}\left(xz\right)}{\overline{\mathbf{G}}\left(x\right)}d\frac{\overline{\mathbf{F}}\left(xz\right)}{\overline{\mathbf{F}}\left(x\right)}.$

By applying Proposition B.1.10 in de Haan and Ferreira (2006) to both
$\overline{\mathbf{F}}$ and $\overline{\mathbf{G}},$ it is easy to check that

$\int_{1}^{\infty}\frac{\overline{\mathbf{G}}\left(xz\right)}{\overline{\mathbf{G}}\left(x\right)}d\frac{\overline{\mathbf{F}}\left(xz\right)}{\overline{\mathbf{F}}\left(x\right)}=-\left(1+o\left(1\right)\right)\gamma/\gamma_{1}.$

On the other hand, since $\overline{\mathbf{F}}$ and $\overline{\mathbf{G}}$
satisfy the aforementioned second-order conditions, then in view of Lemma 3 in
Hua and Joe (2011), there exist two constants $a_{1},a_{2}>0,$ such that
$\overline{\mathbf{F}}\left(x\right)=\left(1+o\left(1\right)\right)a_{1}x^{-1/\gamma_{1}}$
and
$\overline{\mathbf{G}}\left(x\right)=\left(1+o\left(1\right)\right)a_{2}x^{-1/\gamma_{2}},$
as $x\rightarrow\infty.$ Therefore
$\overline{F}\left(x\right)=\left(1+o\left(1\right)\right)d_{1}x^{-1/\gamma}$
with $d_{1}=p^{-1}a_{1}a_{2}\gamma/\gamma_{1}.$ ∎

######  Lemma 6.2.

Under the assumptions of Lemma $\ref{Lemma0},$ we have

$\left(i\right)$
$\lim\limits_{t\rightarrow\infty}C\left(t\right)/\overline{G}\left(t\right)=1.$
---
$\left(ii\right)\text{
}\lim\limits_{t\rightarrow\infty}t^{1/\nu}C\left(\mathbb{U}_{F}\left(t\right)\right)=\infty,$
for each $0<\nu\leq 1.$
$\left(iii\right)$ $\lim_{t\rightarrow\infty}\sup_{x\geq
x_{0}}x^{-1/\gamma\pm\epsilon}\left|\left(t\Lambda\left(x\mathbb{U}_{F}\left(t\right)\right)C\left(x\mathbb{U}_{F}\left(t\right)\right)\right)^{-1}-\left(\gamma/\gamma_{1}\right)x^{1/\gamma}\right|=0,$
for $x_{0}>0$ and any sufficiently small $\epsilon>0.$

###### Proof.

For assertion $\left(i\right),$ write
$C\left(t\right)/\overline{G}\left(t\right)=1-\overline{F}\left(t\right)/\overline{G}\left(t\right)$
and observe that from Lemma 6.1 we have
$\overline{F}\left(t\right)/\overline{G}\left(t\right)=\left(1+o\left(1\right)\right)\left(d_{1}/d_{2}\right)t^{1/\gamma_{2}-1/\gamma}.$
Since $1/\gamma_{2}-1/\gamma<0,$ then
$\overline{F}\left(t\right)/\overline{G}\left(t\right)=o\left(1\right),$ that
is $C\left(t\right)/\overline{G}\left(t\right)=1+o\left(1\right)$ as sought.
For result $\left(ii\right),$ Lemma 6.1 implies that
$\mathbb{U}_{F}\left(t\right)=\left(1+o\left(1\right)\right)\left(d_{1}t\right)^{\gamma}$
(as $t\rightarrow\infty$), it follows that
$C\left(\mathbb{U}_{F}\left(t\right)\right)=\left(1+o\left(1\right)\right)d_{2}\left(d_{1}t\right)^{-\gamma/\gamma_{2}}.$
Since $0<\gamma/\gamma_{2}<1,$ then for every $0<\nu\leq 1,$
$t^{1/\nu}C\left(\mathbb{U}_{F}\left(t\right)\right)\rightarrow\infty$ as
$t\rightarrow\infty.$ To prove $\left(iii\right),$ we first show that

$t\Lambda\left(x\mathbb{U}_{F}\left(t\right)\right)C\left(x\mathbb{U}_{F}\left(t\right)\right)-\left(\gamma_{1}/\gamma\right)x^{-1/\gamma}=O\left(\epsilon\right)x^{-1/\gamma\pm\epsilon}.$
(6.26)

Recalling that $\Lambda\left(x\right)=$
$\int_{x}^{\infty}dF\left(z\right)/C\left(z\right)$ and
$\overline{F}\left(\mathbb{U}_{F}\left(t\right)\right)=1/t,$ we write

$t\Lambda\left(x\mathbb{U}_{F}\left(t\right)\right)C\left(x\mathbb{U}_{F}\left(t\right)\right)=-\frac{C\left(x\mathbb{U}_{F}\left(t\right)\right)}{C\left(\mathbb{U}_{F}\left(t\right)\right)}\int_{x}^{\infty}\frac{C\left(\mathbb{U}_{F}\left(t\right)\right)}{C\left(z\mathbb{U}_{F}\left(t\right)\right)}\frac{d\overline{F}\left(z\mathbb{U}_{F}\left(t\right)\right)}{\overline{F}\left(\mathbb{U}_{F}\left(t\right)\right)}.$
(6.27)

Observe now that
$t\Lambda\left(x\mathbb{U}_{F}\left(t\right)\right)C\left(x\mathbb{U}_{F}\left(t\right)\right)-\dfrac{\gamma_{1}}{\gamma}x^{-1/\gamma}$
may be decomposed into the sum of

$D_{1}\left(s;t\right):=-\left(\frac{C\left(x\mathbb{U}_{F}\left(t\right)\right)}{C\left(\mathbb{U}_{F}\left(t\right)\right)}-x^{-1/\gamma_{2}}\right)\int_{x}^{\infty}\frac{C\left(\mathbb{U}_{F}\left(t\right)\right)}{C\left(z\mathbb{U}_{F}\left(t\right)\right)}\frac{d\overline{F}\left(z\mathbb{U}_{F}\left(t\right)\right)}{\overline{F}\left(\mathbb{U}_{F}\left(t\right)\right)},$
$D_{2}\left(s;t\right):=-x^{-1/\gamma_{2}}\int_{x}^{\infty}\left(\frac{C\left(\mathbb{U}_{F}\left(t\right)\right)}{C\left(z\mathbb{U}_{F}\left(t\right)\right)}-z^{1/\gamma_{2}}\right)\frac{d\overline{F}\left(z\mathbb{U}_{F}\left(t\right)\right)}{\overline{F}\left(\mathbb{U}_{F}\left(t\right)\right)}$

and

$D_{3}\left(s;t\right):=-x^{-1/\gamma_{2}}\int_{x}^{\infty}z^{1/\gamma_{2}}d\left(\frac{\overline{F}\left(z\mathbb{U}_{F}\left(t\right)\right)}{\overline{F}\left(\mathbb{U}_{F}\left(t\right)\right)}-z^{-1/\gamma}\right).$

By applying Proposition B.1.10 in de Haan and Ferreira (2006) to both $C$ and
$\overline{F}$ with integrations by parts, it is easy to verify that

$\left|t\Lambda\left(x\mathbb{U}_{F}\left(t\right)\right)C\left(x\mathbb{U}_{F}\left(t\right)\right)-\left(\gamma_{1}/\gamma\right)x^{-1/\gamma}\right|\leq\epsilon
x^{-1/\gamma\pm\epsilon}.$

Observe now that
$t\Lambda\left(x\mathbb{U}_{F}\left(t\right)\right)C\left(x\mathbb{U}_{F}\left(t\right)\right)-\left(\gamma/\gamma_{1}\right)x^{1/\gamma}$
is equal to

$\left(\left(t\Lambda\left(x\mathbb{U}_{F}\left(t\right)\right)C\left(x\mathbb{U}_{F}\left(t\right)\right)\right)^{-1}\right)^{-1}-\left(\left(\gamma_{1}/\gamma\right)x^{-1/\gamma}\right)^{-1}.$

By using the mean value theorem, the latter equals

$\frac{\left(\gamma_{1}/\gamma\right)x^{-1/\gamma}-t\Lambda\left(x\mathbb{U}_{F}\left(t\right)\right)C\left(x\mathbb{U}_{F}\left(t\right)\right)}{\left(\psi\left(x;t\right)\right)^{2}},$

where $\psi\left(x;t\right)$ is between
$\left(\gamma_{1}/\gamma\right)x^{-1/\gamma}$ and
$t\Lambda\left(x\mathbb{U}_{F}\left(t\right)\right)C\left(x\mathbb{U}_{F}\left(t\right)\right).$
In view of the representation $\left(\ref{formula1}\right)$ and Potter’s
inequalities, applied to $C$ and $\overline{F},$ with an integration by parts,
we get
$t\Lambda\left(x\mathbb{U}_{F}\left(t\right)\right)C\left(x\mathbb{U}_{F}\left(t\right)\right)\geq\left(1-\epsilon\right)x^{-1/\gamma\pm\epsilon}.$
It follows that
$\left(\psi\left(x;t\right)\right)^{2}\geq\left(1-\epsilon\right)^{2}x^{-2/\gamma\pm
2\epsilon}$ and therefore

$\left|t\Lambda\left(x\mathbb{U}_{F}\left(t\right)\right)C\left(x\mathbb{U}_{F}\left(t\right)\right)-\left(\gamma/\gamma_{1}\right)x^{1/\gamma}\right|\leq\left(1-\epsilon\right)^{-2}\epsilon
x^{1/\gamma\pm\epsilon},$

as sought. ∎