Kernel estimation for the tail index of a right-censored Pareto-type
distribution

Abdelhakim Necir${}^{\ast},$ Louiza Soltane

Department of Mathematics, Mohamed Khider University, Biskra, Algeria

Abstract

We introduce a kernel estimator, to the tail index of a right-censored Pareto-
type distribution, that generalizes Worms’s one (Worms and Worms, 2014) in
terms of weight coefficients. Under some regularity conditions, the asymptotic
normality of the proposed estimator is established. In the framework of the
second-order condition, we derive an asymptotically bias-reduced version to
the new estimator. Through a simulation study, we conclude that one of the
main features of the proposed kernel estimator is its smoothness contrary to
Worms’s one, which behaves, rather erratically, as a function of the number of
largest extreme values. As expected, the bias significantly decreases compared
to that of the non-smoothed estimator with however a slight increase in the
mean squared error.

Keywords: asymptotic distributions; heavy-tailed estimation; kernel
estimation; right-censored data.

AMS 2010 Subject Classification: Primary 62G32; 62G30; secondary 60G70; 60F17.

${}^{\text{*}}$Corresponding author<EMAIL_ADDRESS>
E-mail address:
<EMAIL_ADDRESS>(L. Soltane)

##  1\. Introduction

###  1.1. A review of the tail index estimation

Let $X_{1},X_{2},...,X_{n}$ be independent and identically distributed (iid)
of non-negative random variables (rv’s) as $n$ copies of a rv $X,$ defined
over some probability space $\left(\Omega,\mathcal{A},\mathbf{P}\right),$ with
cumulative distribution function (cdf) $F.\ $We assume that the distribution
tail $\overline{F}:=1-F$ is regularly varying at infinity, with index
$\left(-1/\gamma_{1}\right),$ notation:
$\overline{F}\in\mathcal{RV}_{\left(-1/\gamma_{1}\right)},$ that is

$\lim_{t\rightarrow\infty}\frac{\overline{F}\left(tx\right)}{\overline{F}\left(t\right)}=x^{-1/\gamma_{1}},\text{
for any }x>0,$ (1.1)

where $\gamma_{1}>0$ is called the shape parameter or the tail index or the
extreme value index (EVI). It plays a very crucial role in the analysis of
extremes as it governs the thickness of the distribution right-tail. The most
popular estimator of $\gamma_{1}$ is Hill’s estimator Hill (1975) defined by

$\widehat{\gamma}_{1,k}^{\left(H\right)}:=\frac{1}{k}{\displaystyle\sum\limits_{i=1}^{k}}\log\frac{X_{n-i+1:n}}{X_{n-k:n}}={\displaystyle\sum\limits_{i=1}^{k}}\frac{i}{k}\log\frac{X_{n-i+1:n}}{X_{n-i:n}},$
(1.2)

where $X_{1:n}\leq...\leq X_{n:n}$ denote the order statistics pertaining to
the sample $\left(X_{1},...,X_{n}\right)$ and $k=k_{n}$ is an integer sequence
satisfying $1<k<n,$ $k\rightarrow\infty$ and $k/n\rightarrow 0$ as
$n\rightarrow\infty.$ The discrete character and non-stability of Hill’s
estimator present major drawbacks. Indeed, adding a single large-order
statistic in the calculation of the estimator, that is, increasing $k$ by $1,$
may deviate from the true value of the estimate substantially. Thus, the
plotting of this estimator as a function of the upper order statistics often
gives a zig-zag figure (see Figure $\ref{Figure1}).$

Figure 1.1. Ploting both Hill’s (red line) and CDM’s (blue line) tail index
estimators, as function of $k$ upper order statistics, for a Pareto-type
distribution.

To overcome this issue, Csörgő et al. (1985) introduced more general weighs
instead of the natural one $i/k$ that appears in the second formula of
$\widehat{\gamma}_{1,k}^{\left(H\right)},$ to define the following kernel
estimator

$\widehat{\gamma}_{1,k}^{\left(CDM\right)}\left(K\right):=\sum_{i=1}^{k}\frac{i}{k+1}K\left(\frac{i}{k+1}\right)\log\frac{X_{n-i+1:n}}{X_{n-i:n}},$

where $K$ is a kernel function satisfying the following assumptions:

  * •

$\left[A1\right]$ is non increasing and right-continuous on $\mathbb{R}.$

  * •

$\left[A2\right]$ $K\left(s\right)=0$ for $s\notin\left(0,1\right]$ and
$K\left(s\right)\geq 0$ for $s\in\left(0,1\right].$

  * •

$\left[A3\right]$ $\int_{\mathbb{R}}K\left(s\right)ds=1.$

  * •

$\left[A4\right]$ $K$ and its first and second Lebesgue derivatives
$K^{\prime}$ and $K^{\prime\prime}$ are bounded.

The commonly used kernel functions are: the indicator kernel
$K_{1}:=\mathbf{1}\left\\{\left[0,1\right)\right\\},$ the biweight, triweight
and quadweight kernels respectively defined on $0\leq s<1$ by

$K_{2}\left(s\right):=\frac{15}{8}\left(1-s^{2}\right)^{2},\text{
}K_{3}\left(s\right):=\frac{35}{16}\left(1-s^{2}\right)^{3},\text{
}K_{4}\left(s\right):=\frac{315}{128}\left(1-s^{2}\right)^{4},$ (1.3)

and zero elsewhere, where $\mathbf{1}\left\\{A\right\\}$ stands for the
indicator function of a set $A.$ Note that the indicator kernel $K_{1}$
corresponds to the weigh coefficients of a closely related tail index
estimator to Hill’s one $\widehat{\gamma}_{1,k}^{\left(H\right)}.$ The nice
properties of the kernel estimator
$\widehat{\gamma}_{1,k}^{\left(CDM\right)}\left(K\right)$ are the smoothness
and the stability, contrary to Hill’s one which rather exhibits fluctuations
along the range of upper extreme values. Thanks to these features, the exact
choice of $k$ to be used in the kernel estimator becomes not as crucial as
that in Hill’s one (see, e.g., Groeneboom et al., 2003). For an overview of
the kernel estimates of the tail index for complete data, one refers to Hüsler
et al. (2006), Ciuperca and Mercadier (2010), Goregebeur et al. (2010) and
Caeiro and Henriques-Rodrigues (2019) and references therein. Motivated by the
qualities of this estimation method, recently Benchaira et al. (2016) proposed
a kernel estimator of the tail index for randomly truncated data and
established its asymptotic normality. To the best of our knowledge, when the
data are randomly censored, this estimation approach is not yet addressed in
the extreme value literature. In the following section we present a review of
the existing tail index estimators and then propose kernel estimators to
$\gamma_{1}$ for censored data.

##  2\. Review of tail index estimation for censored data

In the analysis of lifetime, reliability or insurance data, the observations
are usually randomly censored. In other words, in many real situations the
variable of interest $X$ is not always available. An appropriate way to model
this matter, is to introduce a non-negative rv $Y,$ called censoring rv,
independent of $X$ and then to consider the rv $Z:=\min\left(X,Y\right)$ and
the indicator variable $\delta:=\mathbf{1}\left\\{X\leq Y\right\\},$ which
determines whether or not $X$ has been observed. The cdf’s of $Y$ and $Z$ will
be denoted by $G$ and $H$ respectively. The analysis of extreme values of
randomly censored data is a new research topic to which Reiss and Thomas
(2007) made a very brief reference, in Section 6.1, as a first step but with
no asymptotic results. Considering Hall’s model (Hall, 1982), Beirlant et al.
(2007) proposed estimators for the EVI and high quantiles and discussed their
asymptotic properties, when the data are censored by a deterministic
threshold. Einmahl et al. (2008) adapted various EVI estimators to the case
where data are censored by a random threshold and proposed a unified method to
establish their asymptotic normality. In this context, the censoring
distribution is assumed to be regularly varying too, that is
$\overline{G}\in\mathcal{RV}_{\left(-1/\gamma_{2}\right)},$ for some
$\gamma_{2}>0.$ By virtue of the independence of $X$ and $Y,$ we have
$\overline{H}\left(x\right)=\overline{F}\left(x\right)\overline{G}\left(x\right)$
and therefore $\overline{H}\in\mathcal{RV}_{\left(-1/\gamma\right)},$ with
$\gamma:=\gamma_{1}\gamma_{2}/\left(\gamma_{1}+\gamma_{2}\right).$ We also
assume that both $\overline{F}$ and $\overline{G}$ satisfy the second-order
condition of regularly varying functions:

$\displaystyle\overline{F}\left(x\right)$
$\displaystyle=C_{1}x^{-1/\gamma_{1}}\left(1+D_{1}x^{-\beta_{1}}\left(1+o\left(1\right)\right)\right),\text{
as }x\rightarrow\infty,$ (2.4) $\displaystyle\overline{G}\left(y\right)$
$\displaystyle=C_{2}y^{-1/\gamma_{2}}\left(1+D_{2}y^{-\beta_{2}}\left(1+o\left(1\right)\right)\right),\text{
as }y\rightarrow\infty,$ (2.5)

where $\beta_{1},\beta_{2},C_{1},C_{2}$ are positive constants and
$D_{1},D_{2}$ are real constants. The parameters
$\tau_{i}:=-\beta_{i}\gamma_{i}<0,$ $i=1,2$ called the second-order parameters
corresponding to cdf’s $F$ and $G$ respectively. This class of cdf’s is known
by Hall’s models which contains the most usual Pareto-type cdf’s, namely Burr,
Fréchet, GEV, GPD, Student, etc. The previous two conditions together imply
that

$\overline{H}\left(z\right)=Cz^{-1/\gamma}\left(1+D_{\ast}z^{-\beta_{\ast}\gamma}\left(1+o\left(1\right)\right)\right),\text{
as }z\rightarrow\infty,$ (2.6)

where $C:=C_{1}C_{2},$ $\beta_{\ast}:=\min\left(\beta_{1},\beta_{2}\right)$
and

$D_{\ast}:=D_{1}\mathbf{1}\left\\{\beta_{1}<\beta_{2}\right\\}+D_{2}\mathbf{1}\left\\{\beta_{1}>\beta_{2}\right\\}+\left(D_{1}+D_{2}\right)\mathbf{1}\left\\{\beta_{1}=\beta_{2}\right\\}.$

Let
$\left\\{\left(Z_{1},\delta_{1}\right),...,\left(Z_{n},\delta_{n}\right)\right\\}$
be a sample from the couple of rv’s $\left(Z,\delta\right)$ and let
$Z_{1:n}\leq...\leq Z_{n:n}$ denote the order statistics pertaining to
$\left(Z_{1},...,Z_{n}\right).$ If we denote the concomitant of the $i$th
order statistic by $\delta_{\left(i\right)}$ (i.e.
$\delta_{\left(i\right)}=\delta_{j}$ if $Z_{i:n}=Z_{j}),$ then the adapted
Hill estimator of the tail index $\gamma_{1}$ is defined by

$\widehat{\gamma}_{1,k}^{\left(EFG\right)}:=\frac{\widehat{\gamma}_{k}^{\left(H\right)}}{\widehat{p}_{k}},$
(2.7)

where $\widehat{\gamma}_{k}^{\left(H\right)}$ is Hill’s estimator of the tail
index $\gamma$ and
$\widehat{p}_{k}:=k^{-1}\sum\limits_{i=1}^{k}\delta_{\left(n-i+1\right)}$
denotes the estimator of asymptotic proportion of non-censored observations in
the tail given by

$p:=\frac{\gamma}{\gamma_{1}}=\frac{\gamma_{2}}{\gamma_{1}+\gamma_{2}},$ (2.8)

which is the limit, as $z\rightarrow\infty,$ of function

$p\left(z\right):=\mathbf{P}\left(\delta=1\mid Z=z\right).$ (2.9)

Asymptotic representations both to $\widehat{\gamma}_{k}^{\left(H\right)}$ and
$\widehat{p}_{k}$ in terms of Brownian bridges processes, given by Brahimi et
al. (2015), leading to the asymptotic normality of
$\widehat{\gamma}_{1,k}^{\left(EFG\right)}$ under the usual second-order
conditions of regularly varying functions. In this context, the estimation of
the conditional tail index is addressed in Ndao et al. (2014), Ndao et al.
(2016), Stupfler (2016) and Goegebeur et al. (2019). Recently Stupfler (2019)
assumed that the censoring and the censored rv’s are dependant and proposed an
estimation procedure to $\gamma_{1}.$

By using a Kaplan-Meier integral Beirlant et al. (2019) proposed in new
estimator to $\gamma_{1}$ defined by

$\widehat{\gamma}_{1,k}^{\left(W\right)}:=\sum_{j=2}^{k}\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\log\frac{Z_{n-j+1:n}}{Z_{n-j:n}},$
(2.10)

where

$\overline{F}_{n}^{KM}\left(t\right):=\left\\{\begin{array}[c]{lc}\prod\limits_{i=1}^{n}\left(1-\dfrac{\delta_{\left(i\right)}}{n-i+1}\right)^{\mathbf{1}\left\\{Z_{i:n}\leq
t\right\\}}&\text{if }t<Z_{n:n}\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
0&\text{otherwise}\end{array}\right.,$

denotes the well-known Kaplan-Meier product-limit estimator (Kaplan and Meier,
1958) of the underlying cdf $F.$ Actually
$\widehat{\gamma}_{1,k}^{\left(W\right)}$ is a slight modification of the tail
index estimator

$\widetilde{\gamma}_{1,k}^{\left(W\right)}:=\sum_{j=1}^{k}\frac{\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\log\frac{Z_{n-j+1:n}}{Z_{n-j:n}},$

first given by Worms and Worms (2014). We showed in Proposition 6.2 that the
increments

$\frac{\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}-\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)},\text{
}j=1,...,k,$

are negligible (in probability) for all large $n,$ this means that the
estimation of
$\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)/\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)$
by
$\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)/\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)$
is well justified. The authors also showed that for a suitable sequence of
integer $k,$ we have

$\sqrt{k}\left(\widehat{\gamma}_{1,k}^{\left(W\right)}-\gamma_{1}\right)\overset{\mathcal{D}}{\rightarrow}\mathcal{N}\left(\lambda
m,p\gamma_{1}^{2}/\left(2p-1\right)\right),\text{ as }n\rightarrow\infty,$
(2.11)

for some real constants $\lambda,$ where
$m:=-\mathbf{1}\left\\{\beta_{1}\leq\beta_{2}\right\\}\gamma^{2}\beta_{1}D_{1}C^{-\gamma\beta_{1}}p^{-1}\left(1+\beta_{1}\gamma/p\right)^{-1},$
provided that $p>1/2$ which is equivalent to $\gamma_{1}<\gamma_{2}.$ In the
real-life applications, the latter assumption may be realizable. Indeed,
Beirlant et al. (2018) gave an application to insurance data and claim that
exhibits heavy-tail and part of these can be considered to satisfy the
$p>1/2.$ Through simulations, Worms and Worms (2014) showed that
$\widehat{\gamma}_{1,k}^{\left(W\right)}$ performs better than the adapted
Hill estimator $\widehat{\gamma}_{1,k}^{\left(EFG\right)},$ in the weak-
censoring case $p>1/2,$ both in term of bias and mean squared error (MSE).
However the estimator exhibits a slightly high bias which is natural when one
deals with Hill-type estimators. Instead of Kaplan-Meier approach, Brahimi et
al. (2016) proposed another asymptotically normal estimator of $\gamma_{1}$
close to $\widehat{\gamma}_{1,k}^{\left(W\right)}$ that is based of the
Nelson-Aalen nonparametric estimator (Nelson, 1972), which seems to have a
slightly lower bias compared with that of
$\widehat{\gamma}_{1,k}^{\left(W\right)}.$ As expected, the asymptotic biases
and variances corresponding to the two estimators meet.

Recently Bladt et al. (2021) proposed the following class of kernel estimators
defined by

$\widehat{\gamma}_{1,k}^{\left(BAB\right)}\left(\mathcal{K}\right):=\frac{1}{k}\sum_{i=1}^{k}\mathcal{K}\left(\frac{i}{k+1},\widehat{p}_{k}\right)\frac{1}{\log\left(\left(k+1\right)/i\right)}\log\frac{Z_{n-i+1:n}}{Z_{n-k:n}},$
(2.12)

where $\widehat{p}_{k}$ is given in $\left(\ref{AH}\right)$ and $\mathcal{K}$
is a positive kernel satisfying
$p\int_{0}^{1}\mathcal{K}\left(s,p\right)du=1,$ for $p\in\left(0,1\right].$
The particular kernel functions used by the authors are:

$\mathcal{K}_{0}\left(s,p\right):=\frac{1}{p}\log\frac{1}{s},\text{
}\mathcal{K}_{1}\left(s,p\right):=s^{p-1},\text{
}\mathcal{K}_{2}\left(s,p\right):=\frac{s^{p-1}-1}{1-p}.$ (2.13)

Since
$\widehat{\gamma}_{1,k}^{\left(BAB\right)}\left(\mathcal{K}_{0}\right)\equiv\widehat{\gamma}_{1,k}^{\left(EFG\right)},$
then this kernel estimator can be viewed as a generalization of$\
\widehat{\gamma}_{1,k}^{\left(EFG\right)},$ in terms of weight coefficients.
As mentioned in their paper, Worms’s estimator
$\widehat{\gamma}_{1,k}^{\left(W\right)}$ does not fall into this framework,
but it simplified version
$\widehat{\gamma}_{1,k}^{\left(BAB\right)}\left(\mathcal{K}\right)$ does. In
their simulation study, Bladt et al. (2021) pointed out that the MSE
characteristics of the estimator
$\widehat{\gamma}_{1,k}^{\left(BAB\right)}\left(\mathcal{K}_{2}\right)$ are
quite comparable to those of $\widehat{\gamma}_{1,k}^{\left(W\right)}.$
Overall, however, $\widehat{\gamma}_{1,k}^{\left(W\right)}$ performs better
$\widehat{\gamma}_{1,k}^{\left(BAB\right)}\left(\mathcal{K}\right)$ in terms
of bias for the three kernel functions. The asymptotic normality of this
estimator is established by considering both weak and strong censoring cases
$\left(\text{i.e. }0<p\leq 1\right).$ We can summarize the features of BAB’s
estimator in two points: its smoothness compared with
$\widehat{\gamma}_{1,k}^{\left(EFG\right)}$ and its asymptotic normality which
is hold for all $0<p\leq 1$ however that of Worms’s one is limited only to the
interval $p>1/2.$

In the following section we introduce a new kernel estimator for the tail
index $\gamma_{1}$ that generalizes $\widehat{\gamma}_{1,k}^{\left(W\right)}$
is the sense that the two estimators coincide for the indicator function
$K_{1}.$ In other terms, this new kernel estimator is a generalization of
CDM’s estimator $\widehat{\gamma}_{1,k}^{\left(CDM\right)}\left(K\right)$ to
the case of censored data.

###  2.1. A new kernel estimator for $\gamma_{1}$

By using Potter’s inequalities, see e.g. Proposition B.1.10 in de Haan and
Ferreira (2006), to the regularly varying function $F$ together with
assumptions $[A1]-[A4],$ Benchaira et al. (2016) showed that

$\lim_{u\rightarrow\infty}\int_{u}^{\infty}g_{K}^{\prime}\left(\frac{\overline{F}\left(x\right)}{\overline{F}\left(u\right)}\right)\log\frac{x}{u}d\frac{F\left(x\right)}{\overline{F}\left(u\right)}=\gamma_{1}\int_{0}^{\infty}K\left(x\right)dx=\gamma_{1},$

where $g_{K}\left(x\right):=xK\left(x\right)$ and $g^{\prime}$ denotes the
Lebesgue derivative of $g.$ Since $\overline{F}$ is continuous, then
$\overline{F}\left(x\right)=\overline{F}\left(x^{-}\right),$ this allows us to
write

$\frac{\overline{F}\left(x\right)}{\overline{F}\left(u\right)}=\theta_{x,u}\frac{\overline{F}\left(x^{-}\right)}{\overline{F}\left(u\right)}+\left(1-\theta_{x,u}\right)\frac{\overline{F}\left(x\right)}{\overline{F}\left(u\right)},\text{
for }x>u,$

for some arbitrary real number $0<\theta_{x,u}<1.$ Next we will see that,
thanks to the mean value theorem, this formula provides us an estimation of
the derivative $g^{\prime}$ in terrmes of an increment of function $g;$ see
equation $\left(\ref{mv}\right)$ below. The notation
$\psi\left(a^{-}\right):=\lim_{x\uparrow a}\psi\left(x\right)$ stands for the
left-limit of a function $\psi\left(t\right)$ as $t$ approaches $a$ from the
left. By letting $u=Z_{n-k:n}$ and substituting $F$ by Kaplan-Meier estimator
$F_{n}^{KM},$ we derive a kernel estimator to the tail index $\gamma_{1}$
defined by

$\widetilde{\gamma}_{1,k}\left(K\right):=\int_{Z_{n-k:n}}^{\infty}g_{K}^{\prime}\left(\mathcal{F}_{n}\left(x\right)\right)\log\left(\frac{x}{Z_{n-k:n}}\right)\frac{dF_{n}^{KM}\left(x\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)},$

where

$\mathcal{F}_{n}\left(x\right):=\theta_{x,n}\frac{\overline{F}_{n}^{KM}\left(x^{-}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}+\left(1-\theta_{x,n}\right)\frac{\overline{F}_{n}^{KM}\left(x\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)},$

and $\theta_{x,n}:=\theta_{x,Z_{n-k:n}}$ $\left(\text{arbitrary}\right).$ To
rewrite the previous integral into a sum form, we use the following crucial
equation: for a given functional $\phi\left(\cdot;F\right),$ we have

$\int\phi\left(x;F\right)dF\left(x\right)=\int\frac{\phi\left(z;F\right)}{\overline{G}\left(z\right)}\left\\{\overline{G}\left(z\right)dF\left(z\right)\right\\}=\int\frac{\phi\left(z;F\right)}{\overline{G}\left(z\right)}dH^{1}\left(z\right),$
(2.14)

where $H^{1}\left(z\right):=\mathbf{P}\left(Z\leq
z,\delta=1\right)=\int_{0}^{z}\overline{G}\left(x\right)dF\left(x\right),$ see
for instance Stute (1995). The empirical counterparts of integrals in
$\left(\ref{eq}\right)$ are

$\int\phi\left(x;F_{n}^{KM}\right)dF_{n}^{KM}\left(x\right)=\int\frac{\phi\left(z;F_{n}^{KM}\right)}{\overline{G}_{n}^{KM}\left(z^{-}\right)}dH_{n}^{1}\left(z\right),$

where

$\overline{G}_{n}^{KM}\left(t\right):=\left\\{\begin{array}[c]{lc}{\displaystyle\prod\limits_{i=1}^{n}}\left(1-\dfrac{1-\delta_{\left(i\right)}}{n-i+1}\right)^{\mathbf{1}\left\\{Z_{i:n}\leq
t\right\\}}&\text{if }t<Z_{n:n}\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
0&\text{otherwise}\end{array}\right.,$

denotes the Kaplan-Meire estimator of cdf $G$ and
$H_{n}^{1}\left(z\right):=n^{-1}\sum\limits_{i=1}^{n}\mathbf{1}\left\\{Z_{i}\leq
z,\text{ }\delta_{i}=1\right\\},$ is the empirical counterpart of the
subdisribution function $H^{1}.$ We used
$\overline{G}_{n}^{KM}\left(t^{-}\right)$ instead of
$\overline{G}_{n}^{KM}\left(t\right)$ to avoid a division by zero, besides
that $\overline{G}_{n}^{KM}\left(Z_{n:n}\right)=0.$ For convenience, we set

$\phi_{n}\left(x;F\right):=\frac{1}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}g_{K}^{\prime}\left(\mathcal{F}_{n}\left(x\right)\right)\log\frac{x}{Z_{n-k:n}}\mathbf{1}\left\\{x>Z_{n-k:n}\right\\}.$

Thus, the kernel estimator $\widetilde{\gamma}_{1,k}\left(K\right)$ may be
rewritten into

$\int_{0}^{\infty}\phi_{n}\left(x;F_{n}^{KM}\right)dF_{n}\left(x\right)=\int_{0}^{\infty}\frac{\phi_{n}\left(z;F_{n}^{KM}\right)}{\overline{G}_{n}^{KM}\left(z^{-}\right)}dH_{n}^{1}\left(z\right),$

which equals

$\displaystyle\sum_{i=1}^{n}\frac{\mathbf{1}\left\\{Z_{i:n}>Z_{n-k:n}\right\\}}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\frac{\delta_{\left(i\right)}}{n\overline{G}_{n}^{KM}\left(Z_{i:n}^{-}\right)}g_{K}^{\prime}\left(\mathcal{F}_{n}\left(Z_{i:n}\right)\right)\log\frac{Z_{i:n}}{Z_{n-k:n}}.$
$\displaystyle=\sum_{i=n-k+1}^{n}\frac{\delta_{\left(i\right)}}{n\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)\overline{G}_{n}^{KM}\left(Z_{i-1:n}\right)}g_{K}^{\prime}\left(\mathcal{F}_{n}\left(Z_{i:n}\right)\right)\log\frac{Z_{i:n}}{Z_{n-k:n}},$

where

$\mathcal{F}_{n}\left(Z_{i:n}\right):=\theta_{i,n}\frac{\overline{F}_{n}^{KM}\left(Z_{i-1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}+\left(1-\theta_{i,n}\right)\frac{\overline{F}_{n}^{KM}\left(Z_{i:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)},$

where $\left(\theta_{i,n}\right)_{1\leq i\leq k}$ is an arbitrary random
sequence. By changing the index of summation $i$ to $n-j+1,$ yields

$\widetilde{\gamma}_{1,k}\left(K\right)=\sum_{j=1}^{k}\frac{\delta_{\left(n-j+1\right)}}{n\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)\overline{G}_{n}^{KM}\left(Z_{n-j:n}\right)}g_{K}^{\prime}\left(\mathcal{F}_{n}\left(Z_{n-j+1:n}\right)\right)\log\frac{Z_{n-j+1:n}}{Z_{n-k:n}}.$

We showed in Proposition 6.1 that

$\frac{\delta_{\left(n-j+1\right)}}{n\overline{G}_{n}^{KM}\left(Z_{n-j:n}\right)}=\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)-\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right),$

therefore

$\displaystyle\widetilde{\gamma}_{1,k}\left(K\right)$
$\displaystyle=\sum_{j=1}^{k}\left\\{\frac{\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}-\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\right\\}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \times
g_{K}^{\prime}\left(\mathcal{F}_{n}\left(Z_{n-j+1:n}\right)\right)\log\frac{Z_{n-j+1:n}}{Z_{n-k:n}}.$

In view of the mean value theorem, we may choose the sequence of constants
$\theta_{j,n}$ so that

$g_{K}^{\prime}\left(\mathcal{F}_{n}\left(Z_{n-j+1:n}\right)\right)=\frac{g_{K}\left(\frac{\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\right)-g_{K}\left(\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\right)}{\frac{\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}-\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}},$
(2.15)

thus

$\widetilde{\gamma}_{1,k}\left(K\right)=\sum_{j=1}^{k}\left\\{g_{K}\left(\frac{\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\right)-g_{K}\left(\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\right)\right\\}\log\frac{Z_{n-j+1:n}}{Z_{n-k:n}}.$
(2.16)

Recall that $g_{K}\left(x\right)=xK\left(x\right)$ and let

$a_{j}:=g_{K}\left(\frac{\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\right)\text{
and }b_{j}=\log\frac{Z_{n-j:n}}{Z_{n-k:n}}.$

By applying Proposition $\ref{Propo3},$ we may rewrite formula
$\left(\ref{gtild}\right)$ into

$\widetilde{\gamma}_{1,k}\left(K\right)=\sum_{j=1}^{k}\frac{\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}K\left(\frac{\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\right)\log\frac{Z_{n-j+1:n}}{Z_{n-j:n}}.$
(2.17)

By using the same modification as made, in Beirlant et al. (2019), to the
original formula of Worms’s estimator
$\widetilde{\gamma}_{1,k}^{\left(W\right)},$ that is substituting
$\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)/\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)$
by
$\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)/\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right),$
we end up to the final form of our new kernel estimator given by

$\widehat{\gamma}_{1,k}\left(K\right):=\sum_{j=2}^{k}\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}K\left(\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\right)\log\frac{Z_{n-j+1:n}}{Z_{n-j:n}}.$
(2.18)

It is obvious that $\widehat{\gamma}_{1,k}\left(K_{1}\right)$ coincides with
Worms’s estimator $\widehat{\gamma}_{1,k}^{\left(W\right)}$ stated in
$\left(\ref{WW}\right).$ For the sake of simplicity, from now on where there
is no conflict, we limit ourselves to writing $\widehat{\gamma}_{1,k}$ instead
of $\widehat{\gamma}_{1,k}\left(K\right).$ Finally, by using the bias-
reduction approach given by Beirlant et al. (2019), we derive an
asymptotically bias-reduced estimator corresponding to
$\widehat{\gamma}_{1,k}$ defined by

$\widehat{\gamma}_{1,k}^{\ast}:=\widehat{\gamma}_{1,k}-\widehat{\rho}\left\\{T_{k}\left(-\widehat{\tau}_{1}/\widehat{\gamma}_{1,k};K\right)-\widehat{\gamma}_{1,k}\widehat{\eta}_{2}\right\\},$
(2.19)

where $\widehat{\tau}_{1}:=-\beta_{1}\widehat{\gamma}_{1,k}$ is a consistent
estimator of the second-order parameter $\tau_{1}:=-\beta_{1}\gamma_{1}$ of
cdf $F$ in $\left(\ref{C1}\right),$

$\widehat{\rho}:=\frac{1}{\widehat{\eta}_{3}/\widehat{\eta}_{2}-\widehat{\eta}_{1}},$
(2.20)
$\widehat{\eta}_{1}:=\int_{0}^{1}s^{-\widehat{\tau}_{1}}\left(1-\widehat{\tau}_{1}\log
s\right)K\left(s\right)ds,$
$\widehat{\eta}_{2}:=\int_{0}^{1}s^{-\widehat{\tau}_{1}}K\left(s\right)ds,\text{
}\widehat{\eta}_{3}:=\int_{0}^{1}\left(s^{-\widehat{\tau}_{1}}-s^{-2\widehat{\tau}_{1}}\right)K\left(s\right)ds$

and

$\begin{array}[c]{l}T_{k}\left(\omega;K\right):=\dfrac{1}{\omega}{\displaystyle\sum\limits_{j=2}^{k}}\dfrac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}K\left(\dfrac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\right)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \
\times\left\\{\left(\dfrac{Z_{n-j:n}}{Z_{n-k:n}}\right)^{-\omega}-\left(\dfrac{Z_{n-j+1:n}}{Z_{n-k:n}}\right)^{-\omega}\right\\},\text{
}\omega>0.\end{array}$ (2.21)

We checked when one substitute $K$ by the indicator kernel function $K_{1},$
$\widehat{\tau}_{1}$ by $-\beta_{1}\widehat{\gamma}_{1,k}$ and then
$\widehat{p}$ by $\widehat{\gamma}/\widehat{\gamma}_{1},$ the kernel reduced-
bias estimator $\widehat{\gamma}_{1,k}^{\ast}$ meets that of Worms’s one
stated in Beirlant et al. (2019) (equation $\left(9\right)).$

To the best of our knowledge, there is no estimator for $\tau_{1},$ however
there is an adaptive estimation method proposed by Beirlant et al. (2018),
which is based on the minimization of the sample variance to the corresponding
bias-reduced estimator of $\gamma_{1}.$ This adaptive estimator is defined by
$\widehat{\tau}_{1}:=\arg\min_{\tau_{1}\in\mathcal{A}}\sum_{k=2}^{n}\left(\widehat{\gamma}_{1,k}^{\ast}-\overline{\widehat{\gamma}}_{1}^{\ast}\right)^{2},$
where
$\overline{\widehat{\gamma}}_{1}^{\ast}:=n^{-1}\sum_{k=2}^{n}\widehat{\gamma}_{1,k}^{\ast}$
and $\mathcal{A}:=\left\\{-0.5-0.1i\right\\}_{0\leq i\leq 25}.$ The rest of
the paper is organized as follows. In Section 2, we present our main result,
namely the asymptotic normality both of $\widehat{\gamma}_{1,k}$ and
$\widehat{\gamma}_{1,k}^{\ast}$ whose proofs are postponed to Section 4. The
finite sample behavior of the proposed estimators is checked by simulation in
Section 3, where a comparison with the already existing ones is made as well.
Finally, some instrumental Propositions and Lemmas are stated in the Appendix.

##  3\. Main results

######  Theorem 3.1.

Assume that both second-order conditions $\left(\ref{C1}\right)$ and
$\left(\ref{C3}\right)$ hold. Let $k=k_{n}$ be a sequence of integer such that
$\sqrt{k}\left(k/n\right)^{\gamma\beta_{\ast}}\rightarrow\lambda,$ and if
$\lambda=0$ that $n=O\left(n^{B}\right)$ for sufficiently large $B>0.$ For a
given kernel function $K$ satisfying assumptions
$\left[A1\right]-\left[A4\right],$ we have
$\sqrt{k}\left(\widehat{\gamma}_{1,k}-\gamma_{1}\right)\overset{\mathcal{D}}{\rightarrow}\mathcal{N}\left(\lambda
m_{K},\sigma_{K}^{2}\right),$ as $n\rightarrow\infty,$ provided that $p>1/2,$
where
$\sigma_{K}^{2}:=\gamma_{1}^{2}\int_{0}^{1}s^{-1/p+1}K^{2}\left(s\right)ds$
and

$m_{K}:=-\mathbf{1}\left\\{\beta_{1}\leq\beta_{2}\right\\}\beta_{1}D_{1}C^{-\gamma\beta_{1}}\gamma_{1}^{2}\int_{0}^{1}s^{\beta_{1}\gamma_{1}}K\left(s\right)ds.$
(3.22)

######  Remark 3.1.

It is clear that $\sigma_{K_{1}}^{2}=p\gamma_{1}^{2}/\left(2p-1\right)$ and

$m_{K_{1}}=-\mathbf{1}\left\\{\beta_{1}\leq\beta_{2}\right\\}\gamma^{2}\beta_{1}D_{1}C^{-\gamma\beta_{1}}p^{-1}\left(1+\beta_{1}\gamma/p\right)^{-1},$

which coincide respectively with the asymptotic variance and the asymptotic
mean, $\lambda m,$ of Worm’s estimator
$\widehat{\gamma}_{1,k}^{\left(W\right)}$ stated in $\left(\ref{naw}\right).$

For the asymptotic normality of the kernel reduced-bias estimator
$\widehat{\gamma}_{1,k}^{\ast},$ we introduce to the following additional
notations:

$\eta_{1}\left(\tau_{1}\right):=\int_{0}^{1}s^{-\tau_{1}}\left(1-\tau_{1}\log
s\right)K\left(s\right)ds,$ (3.23)
$\eta_{2}\left(\tau_{1}\right):=\int_{0}^{1}s^{-\tau_{1}}K\left(s\right)ds,\text{
}\eta_{3}\left(\tau_{1}\right):=\int_{0}^{1}\left(s^{-\tau_{1}}-s^{-2\tau_{1}}\right)K\left(s\right)ds$
(3.24)

and

$\rho\left(\tau_{1}\right):=\left(\eta_{3}\left(\tau_{1}\right)/\eta_{2}\left(\tau_{1}\right)-\eta_{1}\left(\tau_{1}\right)\right)^{-1}.$
(3.25)

It is worth mentioning that by these new notations, we have
$\eta_{i}\left(\widehat{\tau}_{1}\right)\equiv\widehat{\eta}_{i},$ $i=1,2,3$
and $\rho\left(\widehat{\tau}_{1}\right)\equiv\widehat{\rho}.$

######  Theorem 3.2.

Assume that the assumptions of Theorem 3.1 hold, then

$\sqrt{k}\left(\widehat{\gamma}_{1,k}^{\ast}-\gamma_{1}\right)\overset{\mathcal{D}}{\rightarrow}\mathcal{N}\left(0,\sigma_{K}^{\ast
2}\right),\text{ as }n\rightarrow\infty,$

provided that $p>1/2,$ where $\sigma_{K}^{\ast
2}:=p\gamma_{1}^{2}\int_{0}^{1}t^{-1/p+1}\left(\left(1+\eta_{1}\rho\right)-\rho
s^{-\tau_{1}}\right)^{2}K^{2}\left(t\right)dt,$ with
$\eta_{i}:=\eta_{i}\left(\tau_{1}\right),$ $i=1,2,3$ and
$\rho:=\rho\left(\tau_{1}\right).$

###  3.1. Discussion on the asymptotic biases and variances of
$\widehat{\gamma}_{1,k}$ and $\widehat{\gamma}_{1,k}^{\left(W\right)}$

By considering three kernel functions $K_{2},K_{3}$ and $K_{4}$ introduced in
$\left(\ref{Kfun}\right),$ we show that the absolute asymptotic bias of
$\widehat{\gamma}_{1,k}$ is less than that of
$\widehat{\gamma}_{1,k}^{\left(W\right)},$ however the asymptotic variance
behaves opposite. In the other terms $\left|m_{K}\right|<\left|m\right|$ and
$\sigma_{K}^{2}>\sigma^{2}.$ Indeed, let us write

$\frac{\left|m_{K}\right|}{\left|m\right|}=\left(1+\beta_{1}\gamma_{1}\right)\int_{0}^{1}s^{\beta_{1}\gamma_{1}}K\left(s\right)ds=\left(1+t\right)\int_{0}^{1}s^{t}K\left(s\right)ds=:g\left(t\right),$

where $t=\beta_{1}\gamma_{1}>0.$ It is clear from Figure 3.2 that
$g\left(t\right)<1,$ for any $t>0,$ and therefore
$\left|m_{K}\right|<\left|m\right|.$ To compare the two variances, let us
write

$\frac{\sigma_{K}^{2}}{\sigma^{2}}=\left(\frac{p}{2p-1}\right)^{-1}\int_{0}^{1}s^{-1/p+1}K^{2}\left(s\right)ds=:h\left(p\right),$

for $p>1/2.$ The Figure 3.3 shows in turns that $h\left(p\right)>1$ for any
$p>1/2,$ which implies that $\sigma_{K}^{2}>\sigma^{2}.$ From the two figures
we point out that the quadweight kernel provides a better asymptotic bias
compared with other ones, however the asymptotic variance of its corresponding
tail index estimator is the biggest one. Then for a bais-variance trade-off,
we suggest using the triweight kernel function.

Figure 3.2. Plotting the function $g\left(t\right)$ over $t>0,$ for each
kernel $K_{2},K_{3}$ and $K_{4}.$ Figure 3.3. Plotting the function
$h\left(p\right)$ over $\left(1/2,1\right),$ for each kernel $K_{2},K_{3}$ and
$K_{4}.$

###  3.2. The optimal number of upper extremes

For a given kernel $K,$ we seek the optimal number of upper extremes
$k_{K}^{\ast}$ that minimizes the asymptotic MSE which equals
$\sigma_{K}^{2}/k+\left(k/n\right)^{2\gamma\beta_{\ast}}m_{K}^{2}=:\mathcal{M}\left(k\right).$
Explicitly we have

$\displaystyle\mathcal{M}\left(k\right)\begin{tabular}[c]{l}$:=$\end{tabular}\
\frac{\gamma_{1}^{2}}{k}\int_{0}^{1}s^{-1/p+1}K^{2}\left(s\right)ds$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\mathbf{1}\left\\{\beta_{1}\leq\beta_{2}\right\\}\left(k/n\right)^{2\gamma\beta_{1}}\beta_{1}^{2}D_{1}^{2}C^{-2\gamma\beta_{1}}\gamma_{1}^{4}\left\\{\int_{0}^{1}s^{\beta_{1}\gamma_{1}}K\left(s\right)ds\right\\}^{2}.$

Letting $\alpha:=\beta_{1}\gamma$ and
$\mathcal{D}:=\mathbf{1}\left\\{\beta_{1}\leq\beta_{2}\right\\}D_{1}C^{-\alpha}p^{-1},$
we write

$\gamma_{1}^{-2}\mathcal{M}\left(k\right)=\frac{1}{k}\int_{0}^{1}s^{-1/p+1}K^{2}\left(s\right)ds+\left(\frac{k}{n}\right)^{2\alpha}\left(\alpha\mathcal{D}\right)^{2}\left\\{\int_{0}^{1}s^{\alpha/p}K\left(s\right)ds\right\\}^{2}.$

Using similar arguments as used to the proof of Theorem 5 in Csörgő et al.
(1985), we infer that $k_{K}^{\ast}$ minimizing the right-side of the previous
equation is the integer part of
$n^{2\alpha/\left(2\alpha+1\right)}\left\\{2\alpha^{3}\mathcal{D}^{3}\right\\}^{-1/\left(2\alpha+1\right)}\Phi\left(K\right),$
where

$\Phi\left(K\right):=\left\\{\int_{0}^{1}s^{-1/p+1}K^{2}\left(s\right)ds\right\\}^{1/\left(2\alpha+1\right)}\left\\{\int_{0}^{1}s^{\alpha/p}K\left(s\right)ds\right\\}^{-2/\left(2\alpha+1\right)}.$

In particular, for the indicator kernel function
$K_{1}:=\mathbf{1}\left\\{\left[0,1\right)\right\\},$ we have

$\displaystyle\Phi\left(K_{1}\right)$
$\displaystyle=\left\\{\int_{0}^{1}s^{-1/p+1}ds\right\\}^{1/\left(2\alpha+1\right)}\left\\{\int_{0}^{1}s^{\alpha/p}ds\right\\}^{-2/\left(2\alpha+1\right)}$
$\displaystyle=\left(\frac{p}{2p-1}\right)^{1/\left(2\alpha+1\right)}\left(\alpha/p+1\right)^{2/\left(2\alpha+1\right)}.$

Thereby the optimal top $k$ observations used in Worms’s estimator
$\widehat{\gamma}_{1,k}^{\left(W\right)}$ is

$k_{W}^{\ast}=\left[n^{2/\left(2\alpha+1\right)}\left(\frac{2p\alpha^{3}\mathcal{D}^{3}}{2p-1}\right)^{1/\left(2\alpha+1\right)}\left(\alpha/p+1\right)^{2/\left(2\alpha+1\right)}\right].$
(3.26)

Thus the ratio between the two optimal number of extremes is

$\frac{k_{K}^{\ast}}{k_{W}^{\ast}}\sim\frac{\left\\{\int_{0}^{1}s^{-1/p+1}K^{2}\left(s\right)ds\right\\}^{1/\left(2\alpha+1\right)}\left\\{\int_{0}^{1}s^{\alpha/p}K\left(s\right)ds\right\\}^{-2/\left(2\alpha+1\right)}}{\left(\frac{p}{2p-1}\right)^{1/\left(2\alpha+1\right)}\left(\alpha/p+1\right)^{2/\left(2\alpha+1\right)}}.$
(3.27)

Unfortunately, the optimal choice of the number of upper order statistics to
be used in estimation depends mainly on the unknown slowly varying part of the
tail. This fact makes obtaining a practical strategy for minimizing the
asymptotic mean square error through an appropriate choice of $k$ difficult.
There are numerous heuristic methods to select the optimal number of upper
extremes used in the computation of the tail index estimate. An exhaustive
bibliography to this topic is gathered in the nice survey given by Caeiro and
Gomes (2015). Our choice fell on the method of Reiss and Thomas given in Reiss
and Thomas (2007), page $137.$ In this procedure one defines the optimal
sample fraction

$k^{\ast}:=\arg\min_{1<k<n}\frac{1}{k}\sum_{i=1}^{k}i^{\nu}\left|\widehat{\gamma}_{1,i}-\text{median}\left\\{\widehat{\gamma}_{1,1},...,\widehat{\gamma}_{1,k}\right\\}\right|,$

with suitable constant $0\leq\nu\leq 1/2,$ where $\widehat{\gamma}_{1,i}$
corresponds to the kernel estimator of tail index $\gamma_{1},$ based on the
$i$ upper order statistics, of a Pareto-type model. We claim, in our
simulation study below, that $\nu=0.3$ provides better results both in terms
of bias and MSE. This agrees with that was found by Neves and Fraga Alves
(2004) when considering Hill’s estimator in the non-truncation case. We will
use this procedure to select $k^{\ast}$ the optimal numbers of upper order
statistics used in the computation of the all aforementioned estimators.

##  4\. Simulation study

In this section we will perform a simulation study in order to compare the
finite sample behavior of the kernel estimator $\widehat{\gamma}_{1,k},$ given
in $\left(\ref{gtild}\right),$ with the three estimators
$\widehat{\gamma}_{1,k}^{\left(EFG\right)},$
$\widehat{\gamma}_{1,k}^{\left(W\right)}$ and
$\widehat{\gamma}_{1,k}^{\left(BAB\right)}\left(\mathcal{K}\right)$ stated
respectively in $\left(\ref{AH}\right),$ $\left(\ref{WW}\right)$ and
$\left(\ref{BAB}\right).$ We constructed the two estimator
$\widehat{\gamma}_{1,k}$ and
$\widehat{\gamma}_{1,k}^{\left(BAB\right)}\left(\mathcal{K}\right)$ by
selecting the triweight kernel function $K_{3}$ (defined in
$\left(\ref{Kfun}\right)),$ and $\mathcal{K}_{2}$ (given in in
$\left(\ref{Kbis}\right))$ respectively. For the censoring and censored
distributions functions $F$ and $G,$ will be chosen among the following two
models:

  * •

Burr $\left(\zeta,\gamma\right)$ distribution with right-tail function:

$\overline{\mathcal{L}}\left(x\right)=\left(1+x^{1/\zeta}\right)^{-\zeta/\gamma},\text{
}x\geq 0,\text{ }\zeta>0,\text{ }\gamma>0.$

  * •

Fréchet $\left(\gamma\right)$ distribution with right-tail function:

$\overline{\mathcal{L}}\left(x\right)=1-\exp\left(-x^{-1/\gamma}\right),\text{
}x>0,\text{ }\gamma>0.$

For each given distribution, we generate $2000$ random samples of length $n$
$=$ $500$ and plot the four estimators, their corresponding biases and MSE’s
as function of $k=1,...,500.$ We consider four scenarios, namely: a Burr
distribution censored by another Burr distribution (Figure 4.4) a Fréchet
distribution censored by another Fréchet distribution (Figure 4.5), a Burr
distribution censored by a Fréchet distribution (Figure 4.6) and a Fréchet
distribution censored by a Burr distribution (Figure 4.7). In each scenario,
we considered the two censoring schemes, that is the weak censoring
$\left(p>1/2\right)$ and the strong censoring $\left(p<1/2\right).$ The
parametrization of Fréchet and Burr models is made so that it covers both the
two situations $p>1/2$ and $p<1/2.$ In right panels of the four Figures
4.4-$\ref{Figure7},$ the simulation study shows that both kernel estimators
$\widehat{\gamma}_{1,k}$ and $\widehat{\gamma}_{1,k}^{\left(BAB\right)}$
present a smoothness contrary to both estimators
$\widehat{\gamma}_{1,k}^{\left(W\right)}$ and
$\widehat{\gamma}_{1,k}^{\left(EFG\right)},$ which behave erratically a long
the range of the largest extreme values $k.$ In addition, the two kernel
estimators exhibit a stability and alignment with respect to the true value of
the tail index $\gamma_{1}$ over almost the interval. We also point out that,
in terms of stability, $\widehat{\gamma}_{1,k}$ performs better than
$\widehat{\gamma}_{1,k}^{\left(BAB\right)}$ in the strong censoring case. We
notice that $\widehat{\gamma}_{1,k}^{\left(W\right)}$ meets the true value of
the tail index in a single point $k^{\ast}\in\left\\{1,...,500\right\\}$ while
$\widehat{\gamma}_{1,k}^{\left(EFG\right)}$ does not cross the line
$\gamma_{1}$ at any point, but approaches slightly this one on a small
interval of $k.$ From the middle panels (resp. the right panels), in overall,
$\widehat{\gamma}_{1,k}$ performs better than the three other estimators in
terms of bias (resp. MSE) for the strong censoring case $\left(p>1/2\right),$
however $\widehat{\gamma}_{1,k}^{\left(W\right)}$ seems to be slightly better
than $\widehat{\gamma}_{1,k}$ for the weak censoring $\left(p<1/2\right)$ one.

Figure 4.4. Comparison of the estimators (left-panels)
$\widehat{\gamma}_{1,k}^{(EFG)}\left(\text{red line}\right),$
$\widehat{\gamma}_{1,k}^{(W)}\left(\text{blue line}\right),$
$\widehat{\gamma}_{1,k}^{(BAB)}\left(\text{green line}\right),$
$\widehat{\gamma}_{1,k}$ $\left(\text{black line}\right)$ their biases
(middel-panels) and MSE’s (right-panels) for a Burr distribution censored by
another Burr distribution with $p=2/3$ (top-panels) and $p=1/3$ (bottom-
panels) Figure 4.5. Comparison of the estimators (left-panels)
$\widehat{\gamma}_{1,k}^{(EFG)}\left(\text{red line}\right),$
$\widehat{\gamma}_{1,k}^{(W)}\left(\text{blue line}\right),$
$\widehat{\gamma}_{1,k}^{(BAB)}\left(\text{green line}\right),$
$\widehat{\gamma}_{1,k}$ $\left(\text{black line}\right)$ their biases
(middel-panels) and MSE’s (right-panels) for a Fréchet distribution censored
by another Fréchet distribution with $p=2/3$ (top-panels) and $p=1/3$ (bottom-
panels) Figure 4.6. Comparison of the estimators (left-panels)
$\widehat{\gamma}_{1,k}^{(EFG)}\left(\text{red line}\right),$
$\widehat{\gamma}_{1,k}^{(W)}\left(\text{blue line}\right),$
$\widehat{\gamma}_{1,k}^{(BAB)}\left(\text{green line}\right),$
$\widehat{\gamma}_{1,k}$ $\left(\text{black line}\right)$ their biases
(middel-panels) and MSE’s (right-panels) for a Burr distribution censored by
Fréchet distribution with $p=2/3$ (top-panels) and $p=1/3$ (bottom-panels)
Figure 4.7. Comparison of the estimators (left-panels)
$\widehat{\gamma}_{1,k}^{(EFG)}\left(\text{red line}\right),$
$\widehat{\gamma}_{1,k}^{(W)}\left(\text{blue line}\right),$
$\widehat{\gamma}_{1,k}^{(BAB)}\left(\text{green line}\right),$
$\widehat{\gamma}_{1,k}$ $\left(\text{black line}\right)$ their biases
(middel-panels) and MSE’s (right-panels) for a Fréchet distribution censored
by Burr distribution with $p=2/3$ (top-panels) and $p=1/3$ (bottom-panels)

##  5\. Proofs

###  5.1. Proof of Theorem 3.1

We will adapt the proof of Theorem 1 in Beirlant et al. (2019) to the
framework of the kernel estimation. To begin, let us define the following
quantities

$\widehat{RF}_{j}:=\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)},\text{
}RF_{j}:=\frac{\overline{F}\left(Z_{n-j+1:n}\right)}{\overline{F}\left(Z_{n-k:n}\right)},$
$\widehat{RF}_{j}\left(K\right):=\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}K\left(\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\right),\text{
}$

and

$RF_{j}\left(K\right):=\frac{\overline{F}\left(Z_{n-j+1:n}\right)}{\overline{F}\left(Z_{n-k:n}\right)}K\left(\frac{\overline{F}\left(Z_{n-j+1:n}\right)}{\overline{F}\left(Z_{n-k:n}\right)}\right).$

For further use, we set

$\xi_{j}:=j\log\frac{Z_{n-j+1:n}}{Z_{n-j:n}}\text{ and
}E_{j}^{\left(n\right)}:=j\log\frac{Y_{n-j+1:n}}{Y_{n-j:n}},\text{ for
}j=1,...,k,$

where $Y_{1:n}\leq...\leq Y_{n:n}$ be the order statistics pertaining to the
sample $\left(Y_{j}\right)_{1\leq j\leq n}$ of iid standard Pareto rv’s
defined by $Z_{j}=U_{H}\left(Y_{j}\right),$ with

$U_{H}\left(s\right):=\inf\left\\{x,\text{ }H\left(x\right)\geq
1-1/s\right\\},\text{ }s>1,$ (5.28)

stands for the quantile function pertaining to cdf $H.$ Thanks to
$\left(\ref{H}\right),$ we have

$U_{H}\left(s\right)=C^{\gamma}s^{\gamma}\left(1+\gamma
D_{\ast}C^{-\beta_{\ast}\gamma}s^{-\beta_{\ast}\gamma}\left(1+o\left(1\right)\right)\right),\text{
as }s\rightarrow\infty.$ (5.29)

Since $\left\\{\log Y_{j}\right\\}_{1\leq j\leq k}$ are iid standard
exponential rv’s, then the normalized spacings
$\left\\{E_{j}^{\left(n\right)}\right\\}_{1\leq j\leq k}$ are iid standard
exponential rv’s too; see for instance Theorem 4.6.1 in Arnold et al. (2008).
The following approximation, given by Beirlant et al. (2002), will be one of
the basic keys of the proof:

$\xi_{j}=\xi_{j}^{\prime}+R_{j,n}\text{ where
}\xi_{j}^{\prime}:=\left(\gamma+u_{j,k}^{\gamma\beta_{\ast}}b_{n,k}\right)E_{j}^{\left(n\right)},$
(5.30)

with $u_{j,k}:=\frac{j}{k+1}$ and
$b_{n,k}=-\left(1+o_{\mathbf{P}}\left(1\right)\right)\gamma^{2}\beta_{\ast}D_{\ast}C^{-\gamma\beta_{\ast}}\left(\frac{k+1}{n+1}\right)^{\gamma\beta_{\ast}},$
as $n\rightarrow\infty.$ The remainder term $R_{j,n}$ is described in Theorem
2.1 of the aforementioned paper, which satisfies
$\left|\sum_{i=j}^{k}i^{-1}R_{i,n}\right|=o_{\mathbf{P}}\left(b_{n,k}\log\left(\max\left(u_{j,k}^{-1},1\right)\right)\right).$
Next we show that, in the same probability space
$\left(\Omega,\mathcal{A},\mathbf{P}\right),$ there exists a sequence of iid
standard uniform rv’s $\left(U_{i}\right)_{1\leq i\leq k}$ independent to
$\left(E_{i}^{\left(n\right)}\right)_{1\leq i\leq k},$ such that

$\displaystyle\sqrt{k}\left(\widehat{\gamma}_{1,k}-\gamma_{1}\right)-m_{K}\left\\{\sqrt{k}\left(k/n\right)^{\gamma\beta_{\ast}}\right\\}$
$\displaystyle=\gamma\frac{\sqrt{k}}{k+1}\sum_{i=2}^{k}\left\\{u_{i,k}^{-1}\sum_{j=2}^{i}u_{j,k}^{p-1}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\right\\}\mathcal{A}_{i,n}+o_{\mathbf{P}}\left(1\right),$
(5.31)

where $g_{K}\left(s\right)=sK\left(s\right),$
$\mathcal{A}_{i,n}:=p\left(E_{i}^{\left(n\right)}-1\right)-\left(\mathbf{1}\left\\{U_{i}\leq
p\right\\}-p\right)$ and $m_{K}$ is as in $\left(\ref{mk}\right).$ To this
end, let us rewrite formula $\left(\ref{KF}\right)$ into

$\widehat{\gamma}_{1,k}=\sum_{j=2}^{k}\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}K\left(\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\right)\left\\{\log\frac{Z_{n-j+1:n}}{Z_{n-k:n}}-\log\frac{Z_{n-j:n}}{Z_{n-k:n}}\right\\}.$

It is easy to verify that $\widehat{\gamma}_{1,k}-\gamma_{1}$ may be
decomposed into the sum of

$T_{k,n}^{\left(1\right)}:=\sum_{j=2}^{k}\left(\widehat{RF}_{j}\left(K\right)-RF_{j}\left(K\right)\right)\frac{\xi_{j}^{\prime}}{j},\text{
}T_{k,n}^{\left(2\right)}:=\sum_{j=2}^{k}RF_{j}\left(K\right)\frac{\xi_{j}^{\prime}}{j}-\frac{\gamma}{k+1}\sum_{j=2}^{k}K\left(u_{j,k}^{p}\right),$
$T_{k,n}^{\left(3\right)}:=\frac{\gamma}{k+1}\sum_{j=2}^{k}K\left(u_{j,k}^{p}\right)-\gamma_{1}\text{
and
}T_{k,n}^{\left(4\right)}:=\sum_{j=2}^{k}\widehat{RF}_{j}\left(K\right)\frac{R_{j,n}}{j}.$

It is worth mentioning that, by considering the indicator kernel function
$K_{1},$ the last three (remainder) terms $T_{k,n}^{\left(i\right)},$
$i=2,3,4$ coincide with those stated in the beginning of the proof of Theorem
1 in Beirlant et al. (2019). The authors showed that these terms, times
$\sqrt{k},$ tend to zero in probability as $n\rightarrow\infty.$ By deep
reading the proof, we came to the conclusion that by using the assumption
$\left[A4\right]$ on kernel $K$ we end up with
$\sqrt{k}T_{k,n}^{\left(i\right)}=o_{\mathbf{P}}\left(1\right),$ $i=2,3,4$ as
$n\rightarrow\infty,$ as well, that we omit details. This means that
$T_{k,n}^{\left(1\right)}$ is the only term that contributes to the asymptotic
normality of $\widehat{\gamma}_{1,k}.$ Indeed, using Taylor’s expansion of
second-order to this one yields

$\displaystyle T_{k,n}^{\left(1\right)}$
$\displaystyle=\sum_{j=2}^{k}\left(\widehat{RF}_{j}-RF_{j}\right)g_{K}^{\prime}\left(RF_{j}\right)\frac{\xi_{j}^{\prime}}{j}+\frac{1}{2}\sum_{j=2}^{k}\left(\widehat{RF}_{j}-RF_{j}\right)^{2}g_{K}^{\prime\prime}\left(\widetilde{RF}_{j}\right)\frac{\xi_{j}^{\prime}}{j}$
$\displaystyle=\mathcal{T}_{k,n}^{\left(1\right)}+\mathcal{R}_{k,n}^{\left(1\right)},$

where $\widetilde{RF}_{j}$ is a rv between $\widehat{RF}_{j}$ and $RF_{j}.$
From assumption $\left[A4\right],$ the function $g_{K}^{\prime\prime}$ is
bounded, then using similar arguments of the proof in A.3.2 given in Beirlant
et al. (2019), we show that
$\sqrt{k}\mathcal{R}_{k,n}^{\left(1\right)}=o_{\mathbf{P}}\left(1\right),$ as
$n\rightarrow\infty.$ Let us now focus on the term
$\mathcal{T}_{k,n}^{\left(1\right)}$ which may be made into the sum of

$\mathcal{T}_{k,n}^{\left(1,1\right)}:=\sum_{j=2}^{k}\left\\{\log\frac{\widehat{RF}_{j}}{RF_{j}}\right\\}RF_{j}g_{K}^{\prime}\left(RF_{j}\right)\frac{\xi_{j}^{\prime}}{j}$

and

$\mathcal{T}_{k,n}^{\left(1,2\right)}:=\sum_{j=2}^{k}\left\\{-\log\frac{\widehat{RF}_{j}}{RF_{j}}+\left(\frac{\widehat{RF}_{j}}{RF_{j}}-1\right)\right\\}RF_{j}g_{K}^{\prime}\left(RF_{j}\right)\frac{\xi_{j}^{\prime}}{j}.$

It easy to check that
$\log\widehat{RF}_{j}=\sum_{i=j}^{k}\delta_{\left(n-i+1\right)}\log\frac{i-1}{i},$
therefore

$\mathcal{T}_{k,n}^{\left(1,1\right)}=\sum_{j=2}^{k}\left(\sum_{i=j}^{k}\delta_{\left(n-i+1\right)}\log\frac{i-1}{i}-\log
RF_{j}\right)RF_{j}g_{K}^{\prime}\left(RF_{j}\right)\frac{\xi_{j}^{\prime}}{j}.$

Note that $\sum_{i=j}^{k}\xi_{i}/i=\log\left(Z_{n-j+1:n}/Z_{n-k:n}\right)$ and

$\log RF_{j}=\frac{-1}{\gamma_{1}}\sum_{i=j}^{k}\frac{\xi_{i}}{i}+\left(\log
RF_{j}+\frac{1}{\gamma_{1}}\log\frac{Z_{n-j+1:n}}{Z_{n-k:n}}\right),$

it follows that

$\displaystyle\mathcal{T}_{k,n}^{\left(1,1\right)}$
$\displaystyle=\sum_{j=2}^{k}\left(\frac{1}{\gamma_{1}}\sum_{i=j}^{k}\frac{\xi_{i}}{i}+\sum_{i=j}^{k}\delta_{\left(n-i+1\right)}\log\frac{i-1}{i}\right)RF_{j}g_{K}^{\prime}\left(RF_{j}\right)\frac{\xi_{j}^{\prime}}{j}$
$\displaystyle\ \ \ -\sum_{j=2}^{k}\left\\{\log
RF_{j}+\frac{1}{\gamma_{1}}\log\frac{Z_{n-j+1:n}}{Z_{n-k:n}}\right\\}RF_{j}g_{K}^{\prime}\left(RF_{j}\right)\frac{\xi_{j}^{\prime}}{j}.$

Observe that the first term equals

$\sum_{j=2}^{k}\sum_{i=j}^{k}\left(\frac{1}{\gamma_{1}}\frac{\xi_{i}}{i}+\delta_{\left(n-i+1\right)}\log\frac{i-1}{i}\right)RF_{j}g_{K}^{\prime}\left(RF_{j}\right)\frac{\xi_{j}^{\prime}}{j},$

which, by inverting the sums, becomes
$\sum_{i=2}^{k}\left(\frac{1}{\gamma_{1}}\xi_{i}+\delta_{\left(n-i+1\right)}i\log\frac{i-1}{i}\right)S_{i,k},$
where
$S_{i,k}:=\frac{1}{i}\sum_{j=2}^{i}RF_{j}g_{K}^{\prime}\left(RF_{j}\right)\frac{\xi_{j}^{\prime}}{j},$
$j=2,...,k.$ Thereby $\mathcal{T}_{k,n}^{\left(1,1\right)}$ may be rewritten
into

$\begin{array}[c]{cc}{\displaystyle\sum\limits_{i=2}^{k}}\left(\dfrac{1}{\gamma_{1}}\left(\xi_{i}-\gamma\right)+\delta_{\left(n-i+1\right)}i\log\dfrac{i-1}{i}+p\right)S_{i,k}\vskip
6.0pt plus 2.0pt minus 2.0pt&\\\
-{\displaystyle\sum\limits_{j=2}^{k}}\left(\log
RF_{j}+\dfrac{1}{\gamma_{1}}\log\dfrac{Z_{n-j+1:n}}{Z_{n-k:n}}\right)RF_{j}g_{K}^{\prime}\left(RF_{j}\right)\dfrac{\xi_{j}^{\prime}}{j}&=\mathcal{T}_{k,n}^{\left(1,1,1\right)}-\mathcal{T}_{k,n}^{\left(1,1,2\right)}.\end{array}$

We need to the following additional notations:

$c_{i}:=1+i\log\frac{i-1}{i},\text{
}A_{i,n}:=p\left(E_{i}^{\left(n\right)}-1\right)-\left(\delta_{\left(n-i+1\right)}-p\right)\text{
and
}B_{i,n}:=\frac{1}{\gamma_{1}}b_{n,k}u_{i,k}^{\beta_{\ast}\gamma}E_{i}^{\left(n\right)}.$

By adding $\delta_{\left(n-i+1\right)}$ and subtracting it, then by using the
approximation $\left(\ref{ksiprime}\right),$ we rewrite
$\mathcal{T}_{k,n}^{\left(1,1,1\right)}$ into

$\mathcal{T}_{k,n}^{\left(1,1,1\right)}=\sum_{i=2}^{k}A_{i,n}S_{i,k}+\sum_{i=2}^{k}B_{i,n}S_{i,k}+\sum_{i=2}^{k}\delta_{\left(n-i+1\right)}c_{i}S_{i,k}+\gamma_{1}^{-1}\sum_{i=2}^{k}R_{i,n}S_{i,k}.$

Once again, using assumption $\left[A4\right]$ and Proposition (parts c and d)
in Beirlant et al. (2019), we infer that
$\sqrt{k}\sum_{i=2}^{k}\delta_{\left(n-i+1\right)}c_{i}S_{i,k}=o_{\mathbf{P}}\left(1\right)=\sqrt{k}\sum_{i=2}^{k}R_{i,n}S_{i,k},$
as $n\rightarrow\infty.$ Next we show that

$\sqrt{k}\sum_{i=2}^{k}A_{i,n}S_{i,k}=\mathcal{N}\left(0,\gamma_{1}^{2}\int_{0}^{1}t^{-1/p+1}K^{2}\left(t\right)dt\right)+\mathcal{B}_{1,k}+o_{\mathbf{p}}\left(1\right),$
(5.32)

and
$\sqrt{k}\sum_{i=2}^{k}B_{i,n}S_{i,k}=\mathcal{B}_{2,k}+o_{\mathbf{P}}\left(1\right),$
where $\mathcal{B}_{1,k}$ and $\mathcal{B}_{2,k}$ are asymptotic two biases
that we precise later on. Indeed, let us decompose
$\sum_{i=2}^{k}A_{i,n}S_{i,k}$ into the sum of

$\mathcal{I}_{n1}:=\sum_{i=2}^{k}A_{i,n}\left\\{\frac{1}{i}\sum_{j=2}^{i}u_{j,k}^{p}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\frac{\xi_{j}^{\prime}}{j}\right\\},$
$\mathcal{I}_{n2}:=\sum_{i=2}^{k}A_{i,n}\left\\{\frac{1}{i}\sum_{j=2}^{i}\left\\{V_{j,k}^{p}g_{K}^{\prime}\left(V_{j,k}^{p}\right)-u_{j,k}^{p}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\right\\}\frac{\xi_{j}^{\prime}}{j}\right\\}$

and

$\mathcal{I}_{n3}:=\sum_{i=2}^{k}A_{i,n}\left\\{\frac{1}{i}\sum_{j=2}^{i}V_{j,k}^{p}g_{K}^{\prime}\left(V_{j,k}^{p}\right)C_{j,k,0}\frac{\xi_{j}^{\prime}}{j}\right\\},$

where $C_{j,k,0}$ is a sequence of constants defined in assertion
$\left(33\right)$ in Beirlant et al. (2019). By means of Taylor’s expansion to
function $t\rightarrow tg_{K}^{\prime}\left(t\right),$ with assumption
$\left[A4\right],$ and similar arguments as used to terms $I_{n2}$ and
$I_{n3}$ in Beirlant et al. (2019), we infer that
$\sqrt{k}\mathcal{I}_{n2}=o_{\mathbf{P}}\left(1\right)=\sqrt{k}\mathcal{I}_{n3}.$
Recall, from representation $\left(\ref{ksiprime}\right),$ that
$\xi_{j}^{\prime}$ may be rewritten into
$\gamma+\gamma\left(E_{j}^{\left(n\right)}-1\right)+b_{n,k}u_{j,k}^{\gamma\beta_{\ast}}E_{j}^{\left(n\right)},$
this allows us to decompose the first term $\mathcal{I}_{n1}$ into the sum of

$\mathcal{I}_{n1}^{\left(1\right)}:=\gamma\sum_{i=2}^{k}A_{i,n}\left\\{\frac{1}{i}\sum_{j=2}^{i}u_{j,k}^{p}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\frac{1}{j}\right\\}.$
$\mathcal{I}_{n1}^{\left(2\right)}:=\gamma\sum_{i=2}^{k}A_{i,n}\left\\{\frac{1}{i}\sum_{j=2}^{i}u_{j,k}^{p}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\frac{E_{j}^{\left(n\right)}-1}{j}\right\\}$

and

$\mathcal{I}_{n1}^{\left(3\right)}:=b_{n,k}\sum_{i=2}^{k}A_{i,n}\left\\{\frac{1}{i}\sum_{j=2}^{i}u_{j,k}^{p}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\frac{u_{j,k}^{\gamma\beta_{\ast}}}{j}E_{j}^{\left(n\right)}\right\\}.$

Since $g_{K}^{\prime}$ is bounded, then using similar arguments as used to the
terms $I_{1,n}^{\left(2\right)}$ and $I_{1,n}^{\left(3\right)}$ in Beirlant et
al. (2019), we show that
$\sqrt{k}\mathcal{I}_{n1}^{\left(2\right)}=o_{\mathbf{P}}\left(1\right)=\sqrt{k}\mathcal{I}_{n1}^{\left(3\right)}$
as well, that we omit the details. Let us now focus on the first term

$\mathcal{I}_{n1}^{\left(1\right)}=\frac{\gamma}{k+1}\sum_{i=2}^{k}\left\\{\frac{1}{i}\sum_{j=2}^{i}u_{j,k}^{p-1}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\right\\}\left\\{p\left(E_{i}^{\left(n\right)}-1\right)-\left(\delta_{\left(n-i+1\right)}-p\right)\right\\}.$

Einmahl et al. (2008) showed that the sequence of rv’s
$\delta_{\left(i\right)}$ may be approximated by iid Bernoulli rv’s
$\mathbf{1}\left\\{U_{i}\leq p\right\\},$ where $U_{i}$ is a sequence of
standard uniform rv’s which are independent to $E_{i}^{\left(n\right)}.$
Moreover the authors claim that

$\delta_{\left(i\right)}=\mathbf{1}\left\\{U_{i}\leq
p\left(Z_{i:n}\right)\right\\}=\mathbf{1}\left\\{U_{i}\leq p\circ
U_{H}\left(Y_{i:n}\right)\right\\},$

where $p\left(\cdot\right)$ is the function defined in Section $\ref{sec2}.$
In order to use this approximation, let us decompose
$\mathcal{I}_{n1}^{\left(1\right)}$ into the sum of

$\mathcal{L}_{n1}^{\left(1\right)}:=\gamma\frac{\sqrt{k}}{k+1}\sum_{i=2}^{k}\left\\{\frac{1}{i}\sum_{j=2}^{i}u_{j,k}^{p-1}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\right\\}\left(p\left(E_{i}^{\left(n\right)}-1\right)-\left(\mathbf{1}\left\\{U_{n-i+1}\leq
p\right\\}-p\right)\right),$
$\displaystyle\mathcal{L}_{n1}^{\left(2\right)}\begin{array}[c]{c}:=\end{array}-\gamma\frac{\sqrt{k}}{k+1}\sum_{i=2}^{k}\left\\{\frac{1}{i}\sum_{j=2}^{i}u_{j,k}^{p-1}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\right\\}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\times\left(\mathbf{1}\left\\{U_{n-i+1}\leq p\circ
U_{H}\left(n/i\right)\right\\}-\mathbf{1}\left\\{U_{n-i+1}\leq
p\right\\}\right)$

and

$\displaystyle\mathcal{L}_{n1}^{\left(3\right)}\begin{array}[c]{c}:=\end{array}\gamma\frac{\sqrt{k}}{k+1}\sum_{i=2}^{k}\left\\{\frac{1}{i}\sum_{j=2}^{i}u_{j,k}^{p-1}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\right\\}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
\times\left(\mathbf{1}\left\\{U_{n-i+1}\leq p\circ
U_{H}\left(n/i\right)\right\\}-\mathbf{1}\left\\{U_{n-i+1}\leq p\circ
U_{H}\left(Y_{n-i+1:n}\right)\right\\}\right).$

Note that the symbol $f_{1}\circ f_{2}$ sands for the composition of two
functions $f_{1}$ and $f_{2}.$ Note that $\left(U_{n-i+1}\right)_{1\leq i\leq
n}\overset{\mathcal{D}}{=}\left(U_{i}\right)_{1\leq i\leq n}$ and
$\left(E_{i}^{\left(n\right)}\right)_{1\leq i\leq
k}\overset{\mathcal{D}}{=}\left(E_{i}\right)_{1\leq i\leq k},$ where
$\left(E_{i}^{\left(n\right)}\right)_{1\leq i\leq k}$ is a sequence of iid
standard exponential rv’s, then without loss of generality we may write

$\mathcal{L}_{n1}^{\left(1\right)}=\gamma\frac{\sqrt{k}}{k+1}\sum_{i=2}^{k}\left\\{\frac{1}{i}\sum_{j=2}^{i}u_{j,k}^{p-1}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\right\\}\left(p\left(E_{i}-1\right)-\left(\mathbf{1}\left\\{U_{i}\leq
p\right\\}-p\right)\right).$

Observe that this last may be decomposed into the sum of

$\mathcal{L}_{n1}^{\left(1,1\right)}:=\gamma\frac{\sqrt{k}}{k+1}\sum_{i=2}^{k}d_{i,k}\left(g_{K}\right)\left(p\left(E_{i}-1\right)-\left(\mathbf{1}\left\\{U_{i}\leq
p\right\\}-p\right)\right),$

and

$\mathcal{L}_{n1}^{\left(1,2\right)}:=\gamma\frac{\sqrt{k}}{k+1}\sum_{i=2}^{k}\left\\{u_{i,k}^{-1}\int_{0}^{u_{i,k}}s^{p-1}g_{K}^{\prime}\left(s^{p}\right)ds\right\\}\left(p\left(E_{i}-1\right)-\left(\mathbf{1}\left\\{U_{i}\leq
p\right\\}-p\right)\right),$

where
$d_{i,k}\left(g_{K}\right):=\frac{1}{i}\sum_{j=2}^{i}u_{j,k}^{p-1}g_{K}^{\prime}\left(u_{j,k}^{p}\right)-u_{i,k}^{-1}\int_{0}^{u_{i,k}}s^{p-1}g_{K}^{\prime}\left(s^{p}\right)ds.$
Making use of Lemma 6.1 (see the Appendix) and using similar arguments as used
to the term $R_{k,n}$ in the proof of Proposition $1$ (part (a)) in Beirlant
et al. (2019), we show that
$\mathcal{L}_{n1}^{\left(1,1\right)}=o_{\mathbf{P}}\left(1\right),$ that we
omit further details. It is clear that the variance of
$p\left(E_{i}-1\right)-\left(\mathbf{1}\left\\{U_{i}\leq p\right\\}-p\right)$
equals $p^{2}+p\left(1-p\right)=p,$ thus using Lyapunov’s central limit
theorem (for triangular arrays), we get

$\mathcal{L}_{n1}^{\left(1,2\right)}\overset{\mathcal{D}}{\rightarrow}\mathcal{N}\left(0,p\gamma^{2}\int_{0}^{1}s^{-2}\left(\int_{0}^{s}t^{p-1}g_{K}^{\prime}\left(t^{p}\right)dt\right)^{2}ds\right),\text{
as }n\rightarrow\infty.$

By using a change of variables, we readily showed that

$p\gamma^{2}\int_{0}^{1}s^{-2}\left(\int_{0}^{s}t^{p-1}g_{K}^{\prime}\left(t^{p}\right)dt\right)^{2}ds=\gamma_{1}^{2}\int_{0}^{1}t^{-1/p+1}K^{2}\left(t\right)dt.$

Let us consider the second term $\mathcal{L}_{n1}^{\left(2\right)}.$ From
assertion $\left(57\right)$ in Beirlant et al. (2019), we infer that $p\circ
U_{H}\left(n/i\right)-p=p\left(1-p\right)\left(D_{\gamma}\right)_{\ast}\beta_{\ast}C^{-\gamma\beta_{\ast}}\left(i/n\right)^{\gamma\beta_{\ast}}\left(1+o\left(1\right)\right),$
where

$\left(D_{\gamma}\right)_{\ast}:=\gamma_{1}D_{1}\mathbf{1}\left\\{\beta_{1}<\beta_{2}\right\\}-\gamma_{2}D_{2}\mathbf{1}\left\\{\beta_{1}>\beta_{2}\right\\}+\left(\gamma_{1}D_{1}-\gamma_{2}D_{2}\right)\mathbf{1}\left\\{\beta_{1}=\beta_{2}\right\\}.$

It follows that

$\mathcal{L}_{n1}^{\left(2\right)}=-\gamma
p\left(1-p\right)\left(D_{\gamma}\right)_{\ast}\beta_{\ast}C^{-\gamma\beta_{\ast}}\frac{\sqrt{k}}{k+1}\sum_{i=2}^{k}\left\\{i^{-1}\sum_{j=2}^{i}u_{j,k}^{p-1}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\right\\}\left(i/n\right)^{\gamma\beta_{\ast}},$

which may be decomposed into the sum of

$\mathcal{L}_{n1}^{\left(2,1\right)}:=-\gamma
p\left(1-p\right)\left(D_{\gamma}\right)_{\ast}\beta_{\ast}C^{-\gamma\beta_{\ast}}\frac{\sqrt{k}}{k+1}\sum_{i=2}^{k}\left\\{u_{i,k}^{-1}\int_{0}^{u_{i,k}}s^{p-1}g_{K}^{\prime}\left(s^{p}\right)\right\\}\left(i/n\right)^{\gamma\beta_{\ast}}$

and

$\mathcal{L}_{n1}^{\left(2,2\right)}:=-\gamma
p\left(1-p\right)\left(D_{\gamma}\right)_{\ast}\beta_{\ast}C^{-\gamma\beta_{\ast}}\frac{\sqrt{k}}{k+1}\sum_{i=2}^{k}d_{i,k}\left(g_{K}\right)\left(i/n\right)^{\gamma\beta_{\ast}}.$

Observe that
$\mathcal{L}_{n1}^{\left(2,1\right)}=\left\\{\frac{1}{k+1}\sum_{i=2}^{k}\left(u_{i,k}^{-1+\gamma\beta_{\ast}}\int_{0}^{u_{i,k}}s^{p-1}g_{K}^{\prime}\left(s^{p}\right)\right)\right\\}b_{1,k},$
where $b_{1,k}:=-\gamma
p\left(1-p\right)\left(D_{\gamma}\right)_{\ast}\beta_{\ast}C^{-\gamma\beta_{\ast}}\sqrt{k}\left(\frac{k+1}{n}\right)^{\gamma\beta_{\ast}}.$
The quantity between two braces is a Riemann sum, which converges, as
$n\rightarrow\infty,$ to

$\int_{0}^{1}s^{-1+\gamma\beta_{\ast}}\left(\int_{0}^{s}t^{p-1}g_{K}^{\prime}\left(t^{p}\right)dt\right)ds=\frac{1}{p^{2}}\int_{0}^{1}t^{\gamma_{1}\beta_{\ast}}K\left(t\right)dt.$

Recall that
$\sqrt{k}\left(\frac{k}{n}\right)^{\gamma\beta_{\ast}}=O_{\mathbf{P}}\left(1\right),$
then
$\sqrt{k}\left(\frac{k+1}{n}\right)^{\gamma\beta_{\ast}}=\sqrt{k}\left(\frac{k}{n}\right)^{\gamma\beta_{\ast}}+o_{\mathbf{P}}\left(1\right),$
thus
$\mathcal{L}_{n1}^{\left(2,1\right)}=\mathcal{B}_{1,k}+o_{\mathbf{P}}\left(1\right),$
where

$\mathcal{B}_{1,k}:=-\gamma_{1}\left(1-p\right)\left(D_{\gamma}\right)_{\ast}\beta_{\ast}C^{-\gamma\beta_{\ast}}\int_{0}^{1}t^{\gamma_{1}\beta_{\ast}}K\left(t\right)dt\left\\{\sqrt{k}\left(k/n\right)^{\gamma\beta_{\ast}}\right\\}.$
(5.33)

Once again, making use of Lemma $\ref{lemma1},$ we show that

$\mathcal{L}_{n1}^{\left(2,2\right)}=O_{\mathbf{P}}\left(1\right)k^{-1/2+p}\left(k/n\right)^{\gamma\beta_{\ast}}\left\\{k^{-1}\sum_{i=1}^{k}u_{i,k}^{-1+\gamma\beta_{\ast}}\right\\},$

as $n\rightarrow\infty.$ Since
$k^{-1}\sum_{i=1}^{k}u_{i,k}^{-1+\gamma\beta_{\ast}}$ converges to
$\int_{0}^{1}s^{-1+\gamma\beta_{\ast}}ds=1/\gamma\beta_{\ast}$ (Riemann sum)
and both $k^{-1/2+p}$ and $\left(k/n\right)^{\gamma\beta_{\ast}}$ tend to
zero, this means that
$\mathcal{L}_{n1}^{\left(2,2\right)}=o_{\mathbf{P}}\left(1\right),$ thus
$\mathcal{L}_{n1}^{\left(2\right)}=\mathcal{B}_{1,k}+o_{\mathbf{P}}\left(1\right).$
To finish with the term $\mathcal{L}_{n1},$ we will also show that
$\sqrt{k}\mathcal{L}_{n1}^{\left(3\right)}=o_{\mathbf{P}}\left(1\right)$ as
$n\rightarrow\infty.$ Recall that $g_{K}^{\prime}$ is a bounded, then

$\displaystyle\mathcal{L}_{n1}^{\left(3\right)}$
$\displaystyle=O_{\mathbf{P}}\left(1\right)\frac{\sqrt{k}}{k+1}\sum_{i=2}^{k}\left\\{u_{i,k}^{-1}\sum_{j=2}^{i}u_{j,k}^{p-1}\right\\}$
$\displaystyle\times\left|\mathbf{1}\left\\{U_{n-i+1}\leq p\circ
U_{H}\left(Y_{n-i+1:n}\right)\right\\}-\mathbf{1}\left\\{U_{n-i+1}\leq p\circ
U_{H}\left(n/i\right)\right\\}\right|.$

By using similar arguments as used for the term $B_{1,k}^{\left(1\right)}$ in
Beirlant et al. (2019), we show that
$\sqrt{k}\mathcal{L}_{n1}^{\left(3\right)}=o_{\mathbf{P}}\left(1\right),$
therefore we omit details. We now consider the term

$\sum_{i=2}^{k}B_{i,n}S_{i,k}=\frac{1}{\gamma_{1}}b_{n,k}\sum_{i=2}^{k}u_{i,k}^{\beta_{\ast}\gamma}\left(\frac{1}{i}\sum_{j=2}^{i}RF_{j}g_{K}^{\prime}\left(RF_{j}\right)\frac{\xi_{j}^{\prime}}{j}\right)E_{i}^{\left(n\right)}.$

By using similar decomposition as used to the term
$\sum_{i=2}^{k}A_{i,n}S_{i,k},$ we end up with

$\displaystyle\sum_{i=2}^{k}B_{i,n}S_{i,k}$
$\displaystyle=\left(1+o_{\mathbf{P}}\left(1\right)\right)\frac{1}{\gamma_{1}}b_{n,k}\frac{1}{k+1}\sum_{i=2}^{k}u_{i,k}^{\beta_{\ast}\gamma}\left(\frac{1}{i}\sum_{j=2}^{i}u_{j,k}^{p-1}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\xi_{j}^{\prime}\right)E_{i}$
$\displaystyle=\left(1+o_{\mathbf{P}}\left(1\right)\right)pb_{n,k}\left\\{\frac{1}{k+1}\sum_{i=2}^{k}u_{i,k}^{\beta_{\ast}\gamma}\left(\frac{1}{i}\sum_{j=2}^{i}u_{j,k}^{p-1}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\right)E_{i}\right\\}.$

From Lemma 6.2 the previous factor between two braces converges in
probability, as $n\rightarrow\infty,$ to
$p^{-2}\int_{0}^{1}t^{\gamma_{1}\beta_{\ast}}K\left(t\right)dt,$ therefore
$\sqrt{k}\sum_{i=2}^{k}B_{i,n}S_{i,k}=\mathcal{B}_{2,k}+o_{\mathbf{P}}\left(1\right),$
where

$\mathcal{B}_{2,k}:=-\gamma^{2}\beta_{\ast}D_{\ast}C^{-\gamma\beta_{\ast}}p^{-1}\int_{0}^{1}t^{\gamma_{1}\beta_{\ast}}K\left(t\right)dt\left\\{\sqrt{k}\left(k/n\right)^{\gamma\beta_{\ast}}\right\\}.$
(5.34)

In conclusion, we showed that
$\sqrt{k}\mathcal{T}_{k,n}^{\left(1,1,1\right)}=\mathcal{N}\left(0,\sigma_{K}^{2}\right)+\mathcal{B}_{1,k}+\mathcal{B}_{2,k}+o_{\mathbf{P}}\left(1\right).$
Let us now consider the term $\mathcal{T}_{k,n}^{\left(1,1,2\right)}.$
Following the same steps as used in the proof of subsection A.3.3 in Beirlant
et al. (2019), we show that
$\sqrt{k}\mathcal{T}_{k,n}^{\left(1,1,2\right)}=\mathcal{B}_{3,k}+o_{\mathbf{P}}\left(1\right),$
where

$\mathcal{B}_{3,k}:=-\gamma^{2}\beta_{\ast}D_{1}C^{-\gamma\beta_{\ast}}p^{-2}\int_{0}^{1}t^{\gamma_{1}\beta_{\ast}}K\left(t\right)dt\left\\{\sqrt{k}\left(k/n\right)^{\gamma\beta_{\ast}}\right\\}\mathbf{1}\left\\{\beta_{1}\leq\beta_{2}\right\\},$
(5.35)

thereby
$\sqrt{k}\mathcal{T}_{k,n}^{\left(1,1\right)}=\mathcal{N}\left(0,\sigma_{K}^{2}\right)+\mathcal{B}_{1,k}+\mathcal{B}_{2,k}-\mathcal{B}_{3,k}+o_{\mathbf{P}}\left(1\right).$
Using similar arguments as used to the proof given in subsection A.3.2 of the
same paper, we also show that
$\sqrt{k}\mathcal{T}_{k,n}^{\left(1,2\right)}=\mathcal{B}_{4,k}+o_{\mathbf{P}}\left(1\right),$
where

$\mathcal{B}_{4,k}:=-\gamma^{2}\beta_{\ast}\left(D_{\ast}+\frac{D^{\ast}}{p}\right)C^{-\gamma\beta_{\ast}}p^{-1}\int_{0}^{1}t^{\gamma_{1}\beta_{\ast}}K\left(t\right)dt\left\\{\sqrt{k}\left(\frac{k}{n}\right)^{\gamma\beta_{\ast}}\right\\},$
(5.36)

with
$D^{\ast}:=-pD_{\ast}\mathbf{1}\left\\{\beta_{2}<\beta_{1}\right\\}+\left(D_{1}-pD_{\ast}\right)\mathbf{1}\left\\{\beta_{1}\leq\beta_{2}\right\\}.$
To summarize, we showed that

$\sqrt{k}\left(\widehat{\gamma}_{1,k}-\gamma_{1}\right)=\mathcal{L}_{n1}^{\left(1,1\right)}+\mathcal{B}_{k}+o_{\mathbf{P}}\left(1\right),\text{
as }n\rightarrow\infty,$ (5.37)

where
$\mathcal{L}_{n1}^{\left(1,1\right)}\overset{D}{\rightarrow}\mathcal{N}\left(0,\sigma_{K}^{2}\right)$
and
$\mathcal{B}_{k}:=\mathcal{B}_{1,k}+\mathcal{B}_{2,k}-\mathcal{B}_{3,k}+\mathcal{B}_{4,k}.$
Substituting the four biases by their corresponding formulas, we end up with
$\mathcal{B}_{k}=m_{K}\left\\{\sqrt{k}\left(k/n\right)^{\gamma\beta_{\ast}}\right\\},$
where $m_{K}$ is as in Theorem $\ref{Theorem1},$ which completes the proof.

###  5.2. Proof of Theorem 3.2

Note that $\widehat{\tau}_{1}=-\beta_{1}\widehat{\gamma}_{1,k}$ and
$\rho\left(-\beta_{1}\widehat{\gamma}_{1,k}\right)=\widehat{\rho}$ where
$\rho\left(\cdot\right)$ is as in $\left(\ref{rho-t}\right),$ it follows that

$\widehat{\gamma}_{1,k}^{\ast}=\widehat{\gamma}_{1,k}-\rho\left(-\beta_{1}\widehat{\gamma}_{1,k}\right)\left\\{T_{k}\left(\beta_{1};K\right)-\widehat{\gamma}_{1,k}\int_{0}^{1}s^{\beta_{1}\widehat{\gamma}_{1,k}}K\left(s\right)ds\right\\}.$

It is obvious that

$\displaystyle\widehat{\gamma}_{1,k}^{\ast}-\gamma_{1}$
$\displaystyle=\left(\widehat{\gamma}_{1,k}-\gamma_{1}\right)-\rho\left(-\beta_{1}\widehat{\gamma}_{1,k}\right)\left\\{T_{k}\left(\beta_{1};K\right)-\gamma_{1}\int_{0}^{1}s^{\beta_{1}\gamma_{1}}K\left(s\right)ds\right\\}$
$\displaystyle\ \
+\rho\left(-\beta_{1}\widehat{\gamma}_{1,k}\right)\left(\widehat{\gamma}_{1,k}\int_{0}^{1}s^{\beta_{1}\widehat{\gamma}_{1,k}}K\left(s\right)ds-\gamma_{1}\int_{0}^{1}s^{\beta_{1}\gamma_{1}}K\left(s\right)ds\right).$

Using Taylor’s expansion to function $t\rightarrow
t\int_{0}^{1}s^{t}K\left(s\right)ds,$ we get

$\displaystyle\widehat{\gamma}_{1,k}\int_{0}^{1}s^{\beta_{1}\widehat{\gamma}_{1,k}}K\left(s\right)ds-\gamma_{1}\int_{0}^{1}s^{\beta_{1}\gamma_{1}}K\left(s\right)ds$
$\displaystyle=\left(\widehat{\gamma}_{1,k}-\gamma_{1}\right)\int_{0}^{1}s^{\beta_{1}\gamma_{1}}\left(1+\beta_{1}\gamma_{1}\log
s\right)K\left(s\right)ds$
$\displaystyle+\frac{1}{2}\left(\widehat{\gamma}_{1,k}-\gamma_{1}\right)^{2}\beta_{1}\int_{0}^{1}s^{\beta_{1}\widetilde{\gamma}_{1,k}}\left(\log
s\right)\left(\beta_{1}\widetilde{\gamma}_{1,k}\log
s+2\right)K\left(s\right)ds,$

where $\widetilde{\gamma}_{1,k}$ is between $\widehat{\gamma}_{1,k}$ and
$\gamma_{1}.$ Note that
$s^{t}\left|\log^{m}s\right|<\exp\left(-2m/t\right)\leq 1,$ for any $0<s\leq
1,$ $m\geq 0$ and $t>0.$ On the other hand $K$ is bounded on the real line and
$\widehat{\gamma}_{1,k}\overset{\mathbf{P}}{\rightarrow}\gamma_{1},$ then

$\int_{0}^{1}s^{\beta_{1}\widetilde{\gamma}_{1,k}}\left|\left(\log
s\right)\left(\beta_{1}\widetilde{\gamma}_{1,k}\log
s+2\right)\right|K\left(s\right)ds=O_{\mathbf{P}}\left(1\right),\text{ as
}n\rightarrow\infty.$

From Theorem $\ref{Theorem1},$ we deduce that
$\left(\widehat{\gamma}_{1,k}-\gamma_{1}\right)^{2}=O_{\mathbf{P}}\left(k^{-1}\right),$
therefore

$\widehat{\gamma}_{1,k}\int_{0}^{1}s^{\beta_{1}\widehat{\gamma}_{1,k}}K\left(s\right)ds-\gamma_{1}\int_{0}^{1}s^{\beta_{1}\gamma_{1}}K\left(s\right)ds=\eta_{1}\left(\widehat{\gamma}_{1,k}-\gamma_{1}\right)+O_{\mathbf{P}}\left(k^{-1}\right),$

where $\eta_{1}=\eta_{1}\left(\tau_{1}\right)$ is as in
$\left(\ref{eta1}\right).$ To summarize, we showed that

$\displaystyle\widehat{\gamma}_{1,k}^{\ast}-\gamma_{1}$
$\displaystyle=\left(1+\eta_{1}\rho\left(-\beta_{1}\widehat{\gamma}_{1,k}\right)\right)\left(\widehat{\gamma}_{1,k}-\gamma_{1}\right)$
$\displaystyle-\rho\left(-\beta_{1}\widehat{\gamma}_{1,k}\right)\left\\{T_{k}\left(\beta_{1};K\right)-\gamma_{1}\int_{0}^{1}s^{\beta_{1}\gamma_{1}}K\left(s\right)ds\right\\}+O_{\mathbf{P}}\
\left(k^{-1}\right).$

In the proof of Theorem 3.1 (equation $\left(\ref{app-f}\right)),$ we stated
that

$\sqrt{k}\left(\widehat{\gamma}_{1,k}-\gamma_{1}\right)=\gamma\frac{\sqrt{k}}{k+1}\sum_{i=2}^{k}\left\\{u_{i,k}^{-1}\sum_{j=2}^{i}u_{j,k}^{p-1}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\right\\}A_{i,n}+\mathcal{B}_{k}+o_{\mathbf{P}}\left(1\right),$

where $A_{i,n}:=p\left(E_{i}-1\right)-\left(\mathbf{1}\left\\{U_{i}\leq
p\right\\}-p\right)$ and
$\mathcal{B}_{k}:=m_{K}\sqrt{k}\left(k/n\right)^{\gamma\beta_{1}},$ where
$m_{K}:=-\mathbf{1}\left\\{\beta_{1}\leq\beta_{2}\right\\}\beta_{1}D_{1}C^{-\gamma\beta_{1}}\gamma_{1}^{2}\eta_{2},$
with $\eta_{2}=\eta_{2}\left(\tau_{1}\right)$ is as in
$\left(\ref{eta12}\right).$ Next we provide a Gaussian approximation to
$T_{k}\left(\beta_{1};K\right)$ as well. To this end, we will follow similar
steps as used in the proof of Theorem 3.1 as well as that of Theorem 1 in
Beirlant et al. (2019). Let us write

$\left(\frac{Z_{n-j:n}}{Z_{n-k:n}}\right)^{-\beta_{1}}-\left(\frac{Z_{n-j+1:n}}{Z_{n-k:n}}\right)^{-\beta_{1}}=\exp\left(-\beta_{1}\log\frac{Z_{n-j:n}}{Z_{n-k:n}}\right)-\exp\left(-\beta_{1}\log\frac{Z_{n-j+1:n}}{Z_{n-k:n}}\right).$

Once again, by using Taylor’s expansion to function
$t\rightarrow\exp\left(-\beta_{1}t\right),$ yields

$\displaystyle\frac{1}{\beta_{1}}\left\\{\left(\frac{Z_{n-j:n}}{Z_{n-k:n}}\right)^{-\beta_{1}}-\left(\frac{Z_{n-j+1:n}}{Z_{n-k:n}}\right)^{-\beta_{1}}\right\\}$
$\displaystyle=\left(\frac{Z_{n-j+1:n}}{Z_{n-kn}}\right)^{-\beta_{1}}\log\frac{Z_{n-j+1:n}}{Z_{n-j:n}}+\frac{\beta_{1}}{2}\left(\log\frac{Z_{n-j+1:n}}{Z_{n-j:n}}\right)^{2}\left(\frac{\widetilde{Z}_{j:n}}{Z_{n-k:n}}\right)^{-\beta_{1}},$

for some rv’s $\widetilde{Z}_{j:n}$ satisfying
$Z_{n-j:n}\leq\widetilde{Z}_{j:n}\leq$ $Z_{n-j+1:n}.$ Thus
$T_{k}\left(\beta_{1};K\right)$ may be decomposed into the sum of

$\widetilde{T}_{k}\left(\beta_{1};K\right):=\sum_{j=2}^{k}\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}K\left(\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\right)\left(\frac{Z_{n-j+1:n}}{Z_{n-kn}}\right)^{-\beta_{1}}\log\frac{Z_{n-j+1:n}}{Z_{n-j:n}}$

and

$R_{n}:=\frac{\beta_{1}}{2}\sum_{j=2}^{k}\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}K\left(\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\right)\left(\frac{\widetilde{Z}_{j:n}}{Z_{n-k:n}}\right)^{-\beta_{1}}\left(\log\frac{Z_{n-j+1:n}}{Z_{n-j:n}}\right)^{2}.$

Since $K$ is bounded on $\mathbb{R},$ then
$R_{n}=O_{\mathbf{P}}\left(\widetilde{R}_{n}\right),$ where

$\widetilde{R}_{n}:=\frac{\beta_{1}}{2}\sum_{j=2}^{k}\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}\left(\frac{\widetilde{Z}_{j:n}}{Z_{n-k:n}}\right)^{-\beta_{1}}\left(\log\frac{Z_{n-j+1:n}}{Z_{n-j:n}}\right)^{2}.$

The remainder term $\widetilde{R}_{n}$ corresponds to $R_{n}^{\left(1\right)}$
stated in equation $\left(22\right)$ in Beirlant et al. (2019) which is
negligible in the sense that
$\sqrt{k}R_{n}^{\left(1\right)}=o_{\mathbf{P}}\left(1\right)$ thus
$\sqrt{k}R_{n}=o_{\mathbf{P}}\left(1\right)$ as well. We now focus on the
statistic $\widetilde{T}_{k}\left(\beta_{1};K\right)$ which is somewhat
similar to the kernel estimator $\widehat{\gamma}_{1,k}.$ Then using similar
arguments as used in the proof of Theorem 3.1 we provide a Gaussian
approximation to this one without further details. We summarize the result as
follows

$\displaystyle\sqrt{k}\left\\{T_{k}\left(\beta_{1};K\right)-\gamma_{1}\int_{0}^{1}s^{\beta_{1}\gamma_{1}}K\left(s\right)ds\right\\}$
$\displaystyle=\gamma\frac{\sqrt{k}}{k+1}\sum_{i=2}^{k}\left\\{u_{i,k}^{-1}\sum_{j=2}^{i}u_{j,k}^{p-1}g_{K}^{\prime\ast}\left(u_{j,k}^{p}\right)\right\\}A_{i,n}+\mathbb{B}_{k}+o_{\mathbf{P}}\left(1\right),$

where $g_{K}^{\ast}\left(s\right):=s^{\beta_{1}\gamma_{1}+1}K\left(s\right)$
and
$\mathbb{B}_{k}:=-\mathbf{1}\left\\{\beta_{1}\leq\beta_{2}\right\\}\beta_{1}D_{1}C^{-\gamma\beta_{1}}\gamma_{1}^{2}\eta_{3}\sqrt{k}\left(k/n\right)^{\gamma\beta_{1}},$
with $\eta_{3}=\eta_{3}\left(\tau_{1}\right)$ is as in
$\left(\ref{eta12}\right).$ Recall that
$\widehat{\tau}_{1}=-\beta_{1}\widehat{\gamma}_{1,k}$ is a consistent
estimator for $\tau_{1},$ then by means of the convergence dominate theorem,
we easily showed that
$\rho\left(\widehat{\tau}_{1}\right)\overset{\mathbf{P}}{\rightarrow}\rho\left(\tau_{1}\right)=\rho\left(-\beta_{1}\gamma_{1}\right),$
as $n\rightarrow\infty,$ where

$\rho\left(-\beta_{1}\gamma_{1}\right)=\rho\left(\tau_{1}\right)=\frac{\eta_{2}}{\eta_{3}-\eta_{1}\eta_{2}}.$
(5.38)

Thus, in view of the above two Gaussian approximations, we get

$\sqrt{k}\left(\widehat{\gamma}_{1,k}^{\ast}-\gamma_{1}\right)=\gamma\frac{\sqrt{k}}{k+1}\sum_{i=2}^{k}\left\\{u_{i,k}^{-1}\sum_{j=2}^{i}u_{j,k}^{p-1}\varphi\left(u_{j,k}^{p}\right)\right\\}A_{i,n}+\mathbb{B}_{k}^{\ast}+o_{\mathbf{P}}\left(1\right),$
(5.39)

where

$\varphi\left(s\right):=\left(1+\eta_{1}\rho\left(-\beta_{1}\gamma_{1}\right)\right)g_{K}^{\prime}\left(s\right)-\rho\left(-\beta_{1}\gamma_{1}\right)g_{K}^{\prime\ast}\left(s\right),$
(5.40)

and
$\mathbb{B}_{k}^{\ast}:=\left(1+\eta_{1}\rho\left(-\beta_{1}\gamma_{1}\right)\right)\mathcal{B}_{k}-\rho\left(-\beta_{1}\gamma_{1}\right)\mathbb{B}_{k}.$
The objective now is to establish the asymptotic normality of
$\widehat{\gamma}_{1,k}^{\ast}.$ Using similar arguments as used to the term
$\mathcal{L}_{n1}^{\left(1\right)}$ in the proof of Theorem $\ref{Theorem1},$
we also show that the first term in $\left(\ref{GA}\right),$ converges in
distribution to $\mathcal{N}\left(0,\sigma_{K}^{\ast 2}\right),$ where
$\sigma_{K}^{\ast
2}:=p\gamma^{2}\int_{0}^{1}\left(s^{-1}\int_{0}^{s}t^{p-1}\varphi\left(t^{p}\right)dt\right)^{2}ds.$
Using elementary algebra, we obtain

$\sigma_{K}^{\ast
2}=p\gamma_{1}^{2}\int_{0}^{1}t^{-1/p+1}\left(\left(1+\eta_{1}\rho\left(\tau_{1}\right)\right)-\rho\left(\tau_{1}\right)t^{-\tau_{1}}\right)^{2}K^{2}\left(t\right)dt,$

which meets the asymptotic variance stated in Theorem $\ref{Theorem2}.$ The
explicit form of the bias term $\mathbb{B}_{k}^{\ast}$ is

$\left\\{-\mathbf{1}\left\\{\beta_{1}\leq\beta_{2}\right\\}\beta_{1}D_{1}C^{-\gamma\beta_{1}}\gamma_{1}^{2}\right\\}\left\\{\left(1+\eta_{1}\rho\left(\tau_{1}\right)\right)\eta_{2}-\rho\left(\tau_{1}\right)\eta_{3}\right\\}\sqrt{k}\left(k/n\right)^{\gamma\beta_{1}}.$

Substituting $\rho\left(\tau_{1}\right)$ by its expression
$\left(\ref{rho}\right),$ we get
$\left(1+\eta_{1}\rho\left(\tau_{1}\right)\right)\eta_{2}-\rho\left(\tau_{1}\right)\eta_{3}=0,$
it follows that
$\sqrt{k}\left(\widehat{\gamma}_{1,k}^{\ast}-\gamma_{1}\right)\overset{\mathcal{D}}{=}\mathcal{N}\left(0,\sigma_{K}^{\ast
2}\right)+o_{\mathbf{P}}\left(1\right),$ which completes the proof the
Theorem.

Conclusion. We proposed a smoothed (or a kernel) version of Worms’s estimator
(Worms and Worms, 2014) of the tail index of a Pareto-type distribution for
randomly censored data. This estimator is a generalization of the well-known
kernel estimator of the extreme value index for complete data introduced by
Csörgő et al. (1985). The corresponding bias-reduced version of the new kernel
estimator is derived and its asymptotic normality is established. One of the
main features of this estimator is its stability along the interval of the
number $k$ of top extreme values, contrary to Worms’s one which behaves
erratically in a zig-zag way. The simulation study showed that, in the case of
weak censoring, the given estimator overall performed better than the non-
smoothed one in terms of bias and MSE. However, in the case of strong
censoring, the MSE of Worms’s estimator seems to be better.

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##  6\. Appendix

######  Proposition 6.1.

We have

$\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)-\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)=\frac{\delta_{\left(n-j+1\right)}}{n\overline{G}_{n}^{KM}\left(Z_{n-j:n}\right)},\text{
for }j=1,...,k.$

###### Proof.

It is clear that

$\displaystyle\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)-\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)$
$\displaystyle=\prod\limits_{i=1}^{n-j}\left(1-\dfrac{\delta_{\left(i\right)}}{n-i+1}\right)-\prod\limits_{i=1}^{n-j+1}\left(1-\dfrac{\delta_{\left(i\right)}}{n-i+1}\right)$
$\displaystyle=\frac{\delta_{\left(n-j+1\right)}}{j}\prod\limits_{i=1}^{n-j}\left(1-\dfrac{\delta_{\left(i\right)}}{n-i+1}\right),$

thus

$\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)-\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)=\frac{\delta_{\left(n-j+1\right)}}{j}\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right).$
(6.41)

Let $H_{n}\left(z\right):=n^{-1}\sum_{j=1}^{n}\mathbf{1}\left\\{Z_{j}\leq
z\right\\}$ be the empirical cdf pertaining to the sample $Z_{1},...,Z_{n}.$
Since $\overline{H}_{n}\left(Z_{n-j:n}\right)=j/n$ then
$\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)-\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)$
equals

$\frac{\delta_{\left(n-j+1\right)}}{n}\frac{\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)}{\overline{H}_{n}\left(Z_{n-j:n}\right)}=\frac{\delta_{\left(n-j+1\right)}}{n}\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}^{-}\right)}{\overline{H}_{n}\left(Z_{n-j+1:n}^{-}\right)}.$

From assertion $\left(11\right)$ in Shorack and Wellner (1986) (page 295), we
infer that

$\frac{\overline{F}_{n}^{KM}\left(z^{-}\right)}{\overline{H}_{n}\left(z^{-}\right)}=\frac{1}{\overline{G}_{n}^{KM}\left(z^{-}\right)},$

therefore

$\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)-\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)=\frac{\delta_{\left(n-j+1\right)}}{n\overline{G}_{n}^{KM}\left(Z_{n-j+1:n}^{-}\right)}=\frac{\delta_{\left(n-j+1\right)}}{n\overline{G}_{n}^{KM}\left(Z_{n-j:n}\right)},$

as sought. ∎

######  Proposition 6.2.

We have

$\frac{\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}-\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}=O_{\mathbf{P}}\left(k^{-p}\right),$

uniformly on $1\leq j\leq k,$ which tends to zero in probability as
$n\rightarrow\infty.$

###### Proof.

In view of statement $\left(\ref{D}\right),$ we write

$\frac{\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}-\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}=\frac{\delta_{\left(n-j+1\right)}}{j}\frac{\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}.$
(6.42)

Indeed, form Gill (1980) (page 39) and Zhou (1991) (Theorem 2.2) we have

$\frac{\overline{F}_{n}^{KM}\left(x\right)}{\overline{F}\left(x\right)}=O_{\mathbf{P}}\left(1\right)=\frac{\overline{F}\left(x\right)}{\overline{F}_{n}^{KM}\left(x\right)},\text{
}$

uniformly on $x<Z_{n:n}.$ This implies that the right-side of equation
$\left(\ref{RA}\right)$ is stochastically bounded by
$j^{-1}\overline{F}\left(Z_{n-j:n}\right)/\overline{F}\left(Z_{n-k:n}\right),$
uniformly over $1\leq j\leq k.$ Recall that
$\overline{F}\in\mathcal{RV}_{\left(-1/\gamma_{1}\right)},$ then from Potter’s
inequalities B.1.19 in de Haan and Ferreira (2006) (page 367) we have: for
$\epsilon>0$ and $x\geq 1,$ there exists $t_{0}>0,$ such that for $t\geq
t_{0},$
$\left(1-\epsilon\right)x^{-1/\gamma_{1}-\epsilon}<\overline{F}\left(tx\right)/\overline{F}\left(t\right)<\left(1+\epsilon\right)x^{-1/\gamma_{1}+\epsilon}.$
Letting $t=Z_{n-k:n}$ and $x=Z_{n-j:n}/Z_{n-k:n}$ and applying these
inequalities, yields

$\left(1-\epsilon\right)\left(\frac{Z_{n-j:n}}{Z_{n-k:n}}\right)^{-1/\gamma_{1}-\epsilon}<\frac{\overline{F}\left(Z_{n-j:n}\right)}{\overline{F}\left(Z_{n-k:n}\right)}<\left(1+\epsilon\right)\left(\frac{Z_{n-j:n}}{Z_{n-k:n}}\right)^{-1/\gamma_{1}+\epsilon}.$
(6.43)

Since $U_{H}\in\mathcal{RV}_{\left(\gamma\right)},$ then

$\left(1-\epsilon\right)\left(\frac{Y_{n-j:n}}{Y_{n-k:n}}\right)^{\gamma-\epsilon}<\frac{U_{H}\left(Y_{n-j:n}\right)}{U_{H}\left(Y_{n-k:n}\right)}<\left(1+\epsilon\right)\left(\frac{Y_{n-j:n}}{Y_{n-k:n}}\right)^{\gamma+\epsilon},$

where $Y_{1:n}\leq...\leq Y_{n:n}$ are the order statistics already defined in
the beginning of the proof of Theorem $\ref{Theorem1}.$ On the other hand,
from Corollary 2.2.2 in de Haan and Ferreira (2006) (page 41), we infer that
$\left(j/n\right)Y_{n-j:n}\overset{\mathbf{P}}{\rightarrow}1,$ $j=1,...,k,$ as
$n\rightarrow\infty,$ thus by using the previous inequalities, we get

$\frac{Z_{n-j:n}}{Z_{n-k:n}}=\left(1+o_{\mathbf{P}}\left(1\right)\right)\left(\frac{j}{k}\right)^{-\gamma},\text{
as }n\rightarrow\infty.$

Therefore, thanks of $\left(\ref{Finq}\right),$ we have

$\frac{\overline{F}\left(Z_{n-j:n}\right)}{\overline{F}\left(Z_{n-k:n}\right)}=\left(1+o_{\mathbf{P}}\left(1\right)\right)\left(\frac{j}{k}\right)^{p},\text{
}\left(\text{since }p=\gamma/\gamma_{1}\right),$

uniformly over $1\leq j\leq k.$ Then we showed that

$\frac{\overline{F}_{n}^{KM}\left(Z_{n-j:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}-\frac{\overline{F}_{n}^{KM}\left(Z_{n-j+1:n}\right)}{\overline{F}_{n}^{KM}\left(Z_{n-k:n}\right)}=\left(1+o_{\mathbf{P}}\left(1\right)\right)\frac{1}{j}\left(\frac{j}{k}\right)^{p},$

which completes the proof, since
$\frac{1}{j}\left(\frac{j}{k}\right)^{p}=k^{-p}j^{p-1}<k^{-p},$ for all $1\leq
j\leq k$ and $0<p<1.$ ∎

######  Proposition 6.3.

Let $\left(a_{j}\right)_{0\leq j\leq m}$ and $\left(b_{j}\right)_{0\leq j\leq
m}$ be two sequences of real numbers such that $a_{0}=b_{m}=0,$ then
$\sum_{j=1}^{m}\left(a_{j}-a_{j-1}\right)b_{j-1}=\sum_{j=1}^{m}a_{j}\left(b_{j-1}-b_{j}\right).$

###### Proof.

It is straightforward by elementary algebra. ∎

######  Lemma 6.1.

There exists a positive constant $C=C\left(K\right),$ such that

$\left|d_{i,k}\left(g_{K}\right)\right|\leq
Cu_{i,k}^{-1}\left(k+1\right)^{-p},\text{ for all }2\leq i\leq k.$

###### Proof.

Let us write
$d_{i,k}\left(g_{K}\right)=u_{i.k}^{-1}\Delta_{i,k}\left(g_{K}\right),$ where

$\Delta_{i,k}\left(g_{K}\right):=\frac{1}{k+1}\sum_{j=2}^{i}u_{j,k}^{p-1}g_{K}^{\prime}\left(u_{j,k}^{p}\right)-\int_{0}^{u_{i,k}}s^{p-1}g_{K}^{\prime}\left(s^{p}\right)ds.$

It is easy to check that

$\Delta_{i,k}\left(g_{K}\right)=\sum_{j=2}^{i}\int_{u_{j-1,k}}^{u_{j,k}}\left(u_{j,k}^{p-1}g_{K}^{\prime}\left(u_{j,k}^{p}\right)-s^{p-1}g_{K}^{\prime}\left(s^{p}\right)\right)ds-\int_{0}^{u_{1,k}}s^{p-1}g_{K}^{\prime}\left(s^{p}\right)ds.$
(6.44)

For convenience, we set
$h\left(s\right):=sg_{K}^{\prime}\left(s^{\frac{p}{p-1}}\right),$ $0<s<1,$ and
applying the mean value theorem yields
$h\left(u_{j,k}^{p-1}\right)-h\left(s^{p-1}\right)=\left(u_{j,k}^{p-1}-s^{p-1}\right)h^{\prime}\left(s_{j}^{p-1}\right),$
where $s<s_{j}<u_{j,k},$ with
$h^{\prime}\left(s\right)=g_{K}^{\prime}\left(s^{\frac{p}{p-1}}\right)+\frac{p}{p-1}s^{\frac{p}{p-1}}g_{K}^{\prime\prime}\left(s^{\frac{p}{p-1}}\right).$
On the other terms, we have

$u_{j,k}^{p-1}g_{K}^{\prime}\left(u_{j,k}^{p}\right)-s^{p-1}g_{K}^{\prime}\left(s^{p}\right)=\left(u_{j,k}^{p-1}-s^{p-1}\right)h^{\prime}\left(s_{j}^{p-1}\right),$
(6.45)

where
$h^{\prime}\left(s_{j}^{p-1}\right)=g_{K}^{\prime}\left(s_{j}^{p}\right)+\frac{p}{p-1}s_{j}^{p}g_{K}^{\prime\prime}\left(s_{j}^{p}\right).$
From assumption $\left[A4\right],$ both $g_{K}^{\prime}$ and
$g_{K}^{\prime\prime}$ are bounded, this implies that there exist two positive
constants $M_{1}$ and $M_{2},$ such that
$g_{K}^{\prime}\left(u^{p}\right)<M_{1}$ and
$\left|h^{\prime}\left(u^{p-1}\right)\right|<M_{2},$ for all $0<u<1.$ Thus,
combining $\left(\ref{delta}\right)$ and $\left(\ref{mvt}\right),$ yields

$\left|\Delta_{i,k}\left(g_{K}\right)\right|\leq
M\left\\{\widetilde{\Delta}_{i,k}+p^{-1}u_{1,k}^{p}\right\\},$

where
$\widetilde{\Delta}_{i,k}:=\sum_{j=2}^{i}\int_{u_{j-1,k}}^{u_{j,k}}\left|u_{j,k}^{p-1}-s^{p-1}\right|ds$
and $M:=\max\left(M_{1},M_{2}\right)$ that depends on $K.$ Observe now that
$\widetilde{\Delta}_{i,k}=-\sum_{j=2}^{i}\int_{u_{j-1,k}}^{u_{j,k}}\left(u_{j,k}^{p-1}-s^{p-1}\right)ds=-\Delta_{i,k}+p^{-1}u_{1,k}^{p},$
where

$\Delta_{i,k}:=\frac{1}{k+1}\sum_{j=2}^{i}u_{j,k}^{p-1}-\frac{u_{i,k}^{p}}{p}.$

From assertion $(38)$ of Lemma 1 in Beirlant et al. (2019), we infer that
there exists a positive constants $C^{\ast}$ such that $\Delta_{i,k}\leq
C^{\ast}\left(k+1\right)^{-p},$ for all $2\leq i\leq k,$ therefore

$\left|\Delta_{i,k}\left(g_{K}\right)\right|\leq
M\left\\{\frac{C^{\ast}}{\left(k+1\right)^{p}}+\frac{2p^{-1}}{\left(k+1\right)^{p}}\right\\}=\frac{C}{\left(k+1\right)^{p}},$

where $C:=M\left(C^{\ast}+2p^{-1}\right),$ thus
$\left|d_{i,k}\left(g_{K}\right)\right|\leq
Cu_{i,k}^{-1}\left(k+1\right)^{-p}$ as sought. ∎

######  Lemma 6.2.

We have

$\pi_{k}:=\frac{1}{k+1}\sum_{i=2}^{k}u_{i,k}^{\beta_{\ast}\gamma}\left(\frac{1}{i}\sum_{j=2}^{i}u_{j,k}^{p-1}g_{K}^{\prime}\left(u_{j,k}^{p}\right)\right)E_{i}\overset{\mathbf{P}}{\rightarrow}\frac{1}{p^{2}}\int_{0}^{1}t^{\gamma_{1}\beta_{\ast}}K\left(t\right)dt.$

###### Proof.

Observe that $\pi_{k}$ may be decomposed into the sum of

$\pi_{k,1}:=\frac{1}{k+1}\sum_{i=2}^{k}d_{i,k}\left(g_{K}\right)u_{i,k}^{\beta_{\ast}\gamma}E_{i}$

and

$\pi_{k,2}:=\frac{1}{k+1}\sum_{i=2}^{k}\left\\{u_{i,k}^{\beta_{\ast}\gamma-1}\int_{0}^{u_{i,k}}s^{p-1}g_{K}^{\prime}\left(s^{p}\right)ds\right\\}E_{i}.$

Using Lemma $\ref{lemma1},$ we get $\mathbf{E}\left|\pi_{k,1}\right|\leq
C\left(k+1\right)^{-p}\frac{1}{k+1}\sum_{i=1}^{k}u_{i,k}^{\beta_{\ast}\gamma-1},$
then from the Riemann sum, we have
$\frac{1}{k+1}\sum_{i=1}^{k}u_{i,k}^{\beta_{\ast}\gamma-1}$ tends to
$\left(\beta_{\ast}\gamma\right)^{-1},$ and since $k^{-p}\rightarrow 0,$ then
$\pi_{k,1}\overset{\mathbf{P}}{\rightarrow}0,$ as $n\rightarrow\infty.$ On the
other hand, by means of Lyapunov’s central limit theorem for triangular
arrays, we readily show that
$\pi_{k,2}\overset{\mathbf{P}}{\rightarrow}\int_{0}^{1}t^{\beta_{\ast}\gamma-1}\left(\int_{0}^{t}s^{p-1}g_{K}^{\prime}\left(s^{p}\right)ds\right)dt,$
as $n\rightarrow\infty.$ It is easy to verify, making some change of
variables, that this last equals
$p^{-2}\int_{0}^{1}t^{\gamma_{1}\beta_{\ast}}K\left(t\right)dt,$ which
completes the proof. ∎