Estimation of the extreme value index in a censorship framework: asymptotic and finite sample behaviour

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Estimation of the extreme value index in a censorship framework: asymptotic and finite sample behaviour

Jan Beirlant, Julien Worms, Rym Worms

To cite this version:

Jan Beirlant, Julien Worms, Rym Worms. Estimation of the extreme value index in a censorship

framework: asymptotic and finite sample behaviour. Journal of Statistical Planning and Inference,

Elsevier, 2019, 202, pp.31-56. �hal-01768990�

Estimation of the extreme value index in a censorship framework: asymptotic and finite sample behaviour

Jan Beirlant(1) , Julien Worms

(2) & Rym Worms (3)

(1) KU Leuven

Department of Mathematics and Leuven Statistics Research Center, KU Leuven, Belgium, Department of Mathematical Statistics and Actuarial Science, University of the Free State,

South Africa

e-mail : [email protected]

(2) Universit´ e de Versailles-Saint-Quentin-En-Yvelines Laboratoire de Math´ ematiques de Versailles (CNRS UMR 8100),

F-78035 Versailles Cedex, France, e-mail : [email protected]

(3) Universit´ e Paris-Est

Laboratoire d’Analyse et de Math´ ematiques Appliqu´ ees (CNRS UMR 8050),

UPEMLV, UPEC, F-94010, Cr´ eteil, France, e-mail : [email protected]

Abstract

We revisit the estimation of the extreme value index for randomly censored data from a heavy tailed dis- tribution. We introduce a new class of estimators which encompasses earlier proposals given in Worms and Worms (2014) and Beirlant et al. (2018), which were shown to have good bias properties compared with the pseudo maximum likelihood estimator proposed in Beirlant et al. (2007) and Einmahl et al. (2008).

However the asymptotic normality of the type of estimators first proposed in Worms and Worms (2014) was still lacking. We derive an asymptotic representation and the asymptotic normality of the larger class of es- timators and consider their finite sample behaviour. Special attention is paid to the case of heavy censoring, i.e. where the amount of censoring in the tail is at least 50%. We obtain the asymptotic normality with a classical ?

k rate wherek denotes the number of top data used in the estimation, depending on the degree of censoring.

AMS Classification. Primary 62G32 ; Secondary 62N02

Keywords and phrases. Extreme value index. Tail inference. Random censoring. Asymptotic representation.

1Corresponding author J. Worms

1

1. Introduction

Starting from Beirlant et al. (2007), the estimation of the extreme value index in a censorship framework is of growing interest. Suppose we observe a sample ofnindependent couplespZi, δiq1ďiďn where

Zi“minpXi, Ciq and δi “IX_iďC_i.

The i.i.d. samplespX_iqiďnandpC_iqiďn, of respective continuous distribution functionsF andG, are samples from the variable of interest X and of the censoring variable C, measured onnindividual items (insurance claims, hospitalized patients, ...). The variablesXandCare supposed to be independent and, for convenience only, we will suppose in this work that they are non-negative. We will denote by Z_1,n ď . . . ď Z_i,n ď . . .ďZn,n the order statistics associated to the observed sample, and by pδ1,n, . . . , δn,nqthe corresponding indicators of non-censorship.

Einmahl et al. (2008) presented a general method for adapting estimators of the extreme value index in this censorship framework. Worms and Worms (2014) proposed a more survival analysis-oriented approach restricted to the heavy tail case, while Diop et al. (2014) extended the framework to data with covariate information. Beirlant et al. (2016) and Beirlant et al. (2018) proposed bias-reduced versions of two existing estimators. See also Brahimi et al. (2015), Brahimi et al. (2016) and Brahimi et al. (2018) for other papers on the subject.

In this paper, we propose a new class of estimators that encompasses one of the estimators proposed in Worms and Worms (2014) and propose a novel approach to prove the asymptotic normality of these estimators which was unknown up to now for the caseβ“0. We consider here that the distributionsF and Gare heavy-tailed, with positive and respective extreme value indices (EVI)γ₁andγ₂, i.e.

F¯pxq “1´Fpxq “x^´1{γ1lFpxq and Gpyq “¯ 1´Gpyq “y^´1{γ2lGpyq,

where l_F and l_G are slowly varying at infinity. Our target is the EVI γ₁, which we try to recover from our randomly censored observations.

Denoting the distribution function ofZ withH, by independence ofX andC we readily obtain ¯Hpzq “ 1´Hpzq “ z^´1{γl_Hpzq, where l_H “ l_Fl_G and the EVI γ of Z is related to those of X and C via the important relation 1{γ “ 1{γ₁`1{γ<sub>2</sub>. Further in this paper, we will denote by p the crucial quantity p“γ{γ<sub>1</sub> “γ<sub>2</sub>{pγ<sub>1</sub>`γ₂q Ps0,1r, which has to be interpreted as the asymptotic proportion of non-censored observations in the tail.

We assume in this work that the slowly varying functions l_F andl_G satisfy the second order condition first proposed by Hall and Welsh (1985). This yields the so called ”Hall-type” model, i.e. asx, yÑ `8,

F¯pxq “C1x^´1{γ1`

1`D1x<sup>´β1</sup>p1`op1qq˘

(1) Gpyq “¯ C₂y^´1{γ2`

1`D<sub>2</sub>y<sup>´β2</sup>p1`op1qq˘

(2) whereβ1, β2,C1,C2 are positive constants andD1,D2are real constants. Then, setting

C“C1C2, β_˚“minpβ1, β2q, and D_˚ “

$

&

%

D1 if β1ăβ2, D2 if β2ăβ1, D1`D2 if β1“β2, we have, aszÑ 8,

Hpzq “¯ Cz^´1{γ`

1`D<sub>˚</sub>z<sup>´β˚</sup>p1`op1qq˘

. (3)

Correspondingly, withH^´puq “inftz:Hpzq ěuu(0ăuă1) the quantile function corresponding toH, we considerU_Hpxq “H^´p1´1{xq, the right-tail function ofH, for which asxÑ 8,

UHpxq “C^γx^γ`

1`γD<sub>˚</sub>C<sup>´β˚γ</sup>x<sup>´β˚γ</sup>p1`op1qq˘

. (4)

Let us now explain how we build our new family of estimators ofγ₁. For some real numberβ, consider the Box-Cox transformk_´βpuq “şu

1t^´β´1dtforuą1, with the caseβ “0 leading tok₀puq “logpuq. Based on the relation

tÑ8limErk_´βpX{tq |Xąts “ lim

tÑ8

ż8 1

Fputqs

Fsptq dk_´βpuq “ γ₁ 1`βγ1

, (5)

and estimatingFs by the Kaplan-Meier estimatorFs_n^KM defined fortăZ_n,n by Fs_n^KMptq “ ź

Z_i,nďt

ˆ n´i n´i`1

˙δi,n

, (6)

we introduce the following class of statistics Tp_kpβq:“

k

ÿ

j“2

Fs_n^KMpZ_n´j`1,nq Fs_n^KMpZ_n´k,nq

ˆ k_´β

ˆZ_n´j`1,n Zn´k,n

˙

´k_´β

ˆZ_n´j,n Zn´k,n

˙˙

(7) wherek“kndenotes an integer sequence satisfyingknÑ 8andkn “opnq. Withβ“0 we thus obtain the estimator

pγ^pW_1,k^q:“Tpkp0q “

k

ÿ

j“2

Fs_n^KMpZn´j`1,nq Fs_n^KMpZn´k,nq log

ˆZn´j`1,n

Z_n´j,n

˙

(8) of γ1 which was considered in Worms and Worms (2014) and Beirlant et al. (2018). In fact pγ_1,k^pWq turns out to be very close to the estimatorřk

j“1

Fs_n^KMpZ_n´j`1,n^´ q

Fs_n^KMpZ_n´k,nq log^Z_Z^n´j`1,n

n´j,n defined in equation (12) of Worms and Worms (2014) based on ideas issued from the so-called Leurgans approach in survival regression analysis.

The difference concerns a different way to circumvent the use ofFs_n^KM at Zn,n: whether using left-limits or deletingFs_n^KMpZ_n,nqas inpγ^pW_1,k^q.

Note that the statisticsTpkpβqwere used in Beirlant et al. (2018) to obtain a bias-reduced version of the estimatorpγ_1,k^pWq:

pγ_1,k^pBRq“pγ^pW_1,k^q´

p1`β1pγ<sub>1,k</sub><sup>pWq</sup>q<sup>2</sup>p1`2β1pγ_1,k^pWqq pβ₁pγ_1,k^pWqq²

˜

Tp_kpβ₁q ´ pγ_1,k^pWq 1`β₁pγ_1,k^pWq

¸

, (9)

whereβ1denotes the second order parameter ofF in assumption (1).

Now, it is clear from (5) that we can construct the following estimator ofγ1 when the tuning parameter β is supposed to be larger than ´1{γ1:

γp_1,kpβq “ Tp_kpβq

1´βTp_kpβq. (10)

We will compare these estimators with the pseudo maximum likelihood estimator which was first proposed in the random censoring context by Beirlant et al. (2007) and Einmahl et al. (2008):

pγ_1,k^pHq“ 1 ppn

1 k

k

ÿ

i“1

logZ_n´i`1,n Zn´k,n

where pp_n“ 1 k

k

ÿ

i“1

δ_n´i`1,n. (11)

In Beirlant et al. (2018) a small sample simulation study was performed using all those available estimators and it was found thatpγ_1,k^pWqoverall shows quite good bias and MSE performance. However, since no results on the asymptotic normality of this estimator were available yet, these authors proposed the use of a bootstrap algorithm to construct confidence intervals. In this paper we prove the asymptotic normality of pγ1,kpβq in the case p`βγ ą ¹₂. Hence this paper provides the first complete proof of the asymptotic normality forpγ_1,k^pWq in case pą ¹₂, issued from an explicit asymptotic development stated in Theorem 1 of the next section. In the deterministic threshold case, this central limit result (for pγ_1,k^pWq) had already been obtained in Worms and Worms (2018), where a more general competing risks setting was considered, and using a different approach from the present proof.

The restriction pą ¹₂ is rather restrictive for instance in insurance problems such as those discussed in Beirlant et al. (2018) where heavy censoring appears. The introduction of the class of estimators pγ_1,kpβq helps to circumvent this problem when consideringβ ą0.

Finally, in the next section, we will see that our results also lead to the statement of the asymptotic normality of the bias-reduced estimatorpγ^pBRq_1,k , which was not known so far.

Our paper is organized as follows: in Section 2, we state and discuss the asymptotic normality result 3

for pγ1,kpβq and pγ_1,k^pBRq. Section 3 is devoted to the proof. Technical aspects of the proof are postponed to the Appendix. In Section 4 we discuss the finite sample behavior of the different estimators pγ1,kpβq with βą ´1{γ₁, and ofpγ_1,k^pBRq.

2. Results

Our first and main result states the asymptotic behavior of the statisticsTpkpβqdefined in (7). This result entails the asymptotic normality of the estimatorpγ_1,k^pWq ofγ1 by considering the particular caseβ“0. The main condition is that the heaviness of the tail of the censoring variableC should be sufficiently high with respect to the one of the variableX. More precisely, introducing the notationpβ“p`γβ“pp1`γ1βq, the condition is be thatpβ must be larger than 1{2 (i.e. γ2ąγ1{p1`2γ1βq).

Theorem 1. Let conditionsp1qandp2qhold. We assume further thatp_βą¹₂, and

?

kpk{nq^{γβ˚ nÑ8}ÝÑ λ, (12)

and, ifλ“0, that n“Opk^B_nqfor some large enoughBą0. We then have, asnÑ 8,

?k ˆ

Tp_kpβq ´ γ₁ 1`γ1β

˙

“G_n`λm<sub>β</sub>`o_Pp1q where G_n“^d γ pβ

?1 k

k

ÿ

i“2

u^p_i,k^β´1pppE_i´1q ´ pIUiďp´pqq with pEiq and pUiq denoting independent iid samples with, respectively, standard exponential and standard uniform distributions, and

m_β“

"

´γ²β1D1C^´γβ1p^´1_β ppβ`γβ1q^´1 ifβ1ďβ2,

0 ifβ1ąβ2.

Therefore, asnÑ 8,

? k

ˆ

Tpkpβq ´ γ1

1`γ₁β

˙

ÝÑd Npλmβ, σ²_βq where σ²_β“ γ² p²_β

p

2p_β´1 “γ₁² p 2p´1

p² p²_β

2p´1 2p_β´1.

Since Gn is a sum of independent random variables, it is then easy, using Lyapunov’s CLT and the delta-method, to derive the following asymptotic normality result for the family of estimatorspγ_1,kpβqof γ₁ defined by (10).

Corollary 1. Under the conditions of Theorem 1, asnÑ 8,

?

kppγ_1,kpβq ´γ₁qÝÑ^d Npλm_γ1,β, σ²_γ

1,βq where

σ_γ²

1,β “ γ² p²_β

p

2pβ´1p1`βγ₁q⁴“γ²₁ p

2p´1p1`βγ₁q² 2p´1 2pβ´1 and

mγ₁,β “

"

´γ²β1D1C^´γβ1p^´1_β ppβ`γβ1q<sup>´1</sup>p1`βγ1q² if β1ďβ2,

0 if β1ąβ2.

Remark 1. Since pγ_1,k^pWq “ Tp_kp0q “ pγ_1,kp0q, taking β “ 0 in Theorem 1 or in Corollary 1 entails the asymptotic normality for pγ_1,k^pWq when pą 1{2, i.e. when γ2 ąγ1. When β ą0, the asymptotic normality forpγ1,kpβqholds under the weaker assumption pβą1{2, i.e. γ2ąγ1{p1`2γ1βq, and therefore allowing for stronger censoring in the tail. On the other hand the restriction becomes worse for negativeβ.

Whenβ₁ďβ₂ the absolute value of the asymptotic bias of pγ_1,kpβqis increasing inβ. For a bias comparison for the caseβ1ąβ2one needs third order assumptions. On the other hand the asymptotic variance ofpγ1,kpβq is decreasing in β as long as pβ ă1 and is increasing as pβ ą1. It is difficult to say anything in general about the comparison of the asymptotic mean-squared error of pγ_1,kpβq with respect toγp_1,k^pWq. It is of course, whenβ ą0 andpgets close to the value1{2, in favor ofpγ_1,kpβq, at least from a theoretical point of view.

Remark 2. From Einmahl et al. (2008) it follows that the asymptotic variance of pγ_1,k^pHq is given by ¹_k^γ_p¹², which, for all1{2ăpă1is lower than the asymptotic variance ¹_k2p´1^pγ21 of pγ_1,k^pWq.

On the other hand, in case β₁ ďβ₂ it follows from Beirlant et al. (2016) that the absolute value of the asymptotic bias of pγ_1,k^pHq equals pk{nq^γβ˚|m_γ1,0|<sup>1`γ</sup><sub>1`γβ</sub>^1β1

1 , which is larger than pk{nq^γβ˚|m_γ1,0| stated in the above theorem.

Remark 3. The asymptotic distribution of pγ_1,k^pWq in casepď ¹₂, and in general of pγ1,kpβqin case pβ ď ¹₂, is not known. The authors conjecture that asymptotic normality still holds, however with a slower rate than k^´1{2, presumablyk^´p whenpă1{2, but the method of proof outlined below could not be carried through in that case.

Combining the asymptotic developments of γp_1,k^pWqand Tpkpβq forβ “β1, which are both weighted sums of the same i.i.d. random variablesppEi´1q ´ pIUiďp´pq, and relying on the two-dimensional Lyapunov’s CLT and the delta-method, it is now possible to deduce the following asymptotic normality result for the bias-reduced version ofpγ_1,k^pWq introduced in Beirlant et al. (2018). The proof is omitted for brevity.

Corollary 2. Under the conditions of Theorem 1 and assuming that pą1{2, asnÑ 8, we have

?

kppγ^pBRq_1,k ´γ₁qÝÑ^d Np0, σ_pBRq² q where, withδ“pβ₁´p“γβ1,

σ_pBRq² :“γ₁² p 2p´1

pp`δq<sup>2</sup>ppp`δq²` p1´pq<sup>2</sup>`δ`δ<sup>2</sup>q δ<sup>2</sup>p2p´1`δqp2p´1`2δq .

Remark 4. While the asymptotic bias ofpγ_1,k^pBRqis always0, its asymptotic variance is in general larger than those of the competing estimators.

3. Proof of Theorem 1

Let us introduce the following important notations with 1ďi, jďk:

ξj“jlogZn´j`1,n

Zn´j,n

and ui,k“ i

k`1, (13)

as well as the ratios

RFyj “Fs_n^KMpZn´j`1,nq

Fs_n^KMpZn´k,nq and RFj“ FpZs n´j`1,nq

FpZs n´k,nq . (14)

If we also defineξj,k,β“ξj ifβ“0 and otherwise ξ_j,k,β “ j

ˆ k_´β

ˆZn´j`1,n

Z_n´k,n

˙

´k_´β

ˆZn´j,n

Z_n´k,n

˙˙

“ j β

˜ˆ Zn´j,n

Z_n´k,n

˙´β

´

ˆZn´j`1,n

Z_n´k,n

˙´β¸

then, fromp7q, we have

Tpkpβq:“

k

ÿ

j“2

Fs_n^KMpZn´j`1,nq Fs_n^KMpZn´k,nq

ξj,k,β

j where, using a Taylor expansion (of order 2) ,

ξj,k,β “ j β

ˆ

exp^´βlog

´_Zn´j,n

Zn´k,n

¯

´exp^´βlog

´_Zn´j`1,n

Zn´k,n

¯˙

“ ξj

ˆZ_n´j`1,n Zn´k,n

˙´β

`βξ²_j 2j

˜ Z˜_j,n Zn´k,n

¸´β

, (15) for some variables ˜Zj,n satisfyingZn´j,nďZ˜j,nďZn´j`1,n.

The overall objective is to appropriately use the relation between the variablesξ_jand standard exponential order statisticsE_j^pnqdefined below, as well as between the ratiosRFjandpZn´j`1,n{Zn´k,nq^´β and uniform order statisticsVj,k(with meanuj,k) also defined below, in order to prove Theorem 1. Indeed, letpYiqdenote i.i.d. standard Pareto rv’s defined byZi“UHpYiq, and let

Y˜_{k´j`1,k</sub>“Y<sub>n´j`1,n}{Y_n´k,n, V_j,k“1{Y˜_{k´j`1,k</sub>, and E<sub>j</sub><sup>pnq</sup>“jlogpY<sub>n´j`1,n}{Y_n´j,nq, 1ďj ďk. (16) 5

It is then known that pV_1,k, . . . , V_j,k, . . . , V_k,kq follows the distribution of the vector of order statistics of a standard uniform random sample of size k, and that the variables pE₁^pnq, . . . , E_k^pnqq are jointly equal in distribution to a sample of sizek of independent standard exponential rv’s.

Beirlant et. al. (2002) showed that the rv’sξ_j andE_j^pnqare related as follows:

ξj “ξ_j¹`Rn,j, where we defineξ<sup>1</sup><sub>j</sub>“ pγ`u^γβ_j,k^˚bn,kqE_j^pnq, (17) whereb_n,k is asymptotically equivalent to´γ²β_˚D_˚C^´γβ˚´

k`1 n`1

¯γβ_˚

, ask, nÑ 8andk{nÑ0. Properties of the remainder termRn,j will be detailed in Subsection 3.1 . Equationp17qthus implies that

ξ_j,k,β “ξ_j,k,β¹ `R_n,j,β, (18)

where

ξ_j,k,β¹ “ ξ_j¹

ˆZn´j`1,n

Z_n´k,n

˙´β

(19)

Rn,j,β “ Rn,j

ˆZn´j`1,n

Z_n´k,n

˙´β

`βξ²_j 2j

˜ Z˜j,n

Z_n´k,n

¸´β

. (20)

We can now start breaking downTp_kpβq ´_1`γ^γ1

1β into several terms by writing:

Tpkpβq ´ γ₁ 1`γ1β “

k

ÿ

j“2

RFyj

ξ_j,k,β

j ´ γ₁ 1`γ1β “

˜ _k ÿ

j“2

RFyj

ξ_j,k,β¹

j ´ γ₁ 1`γ1β

¸

`

k

ÿ

j“2

yRFj

R_n,j,β j

“

k

ÿ

j“2

˜ RFy_j

RFj

´1

¸ RFj

ξ_j,k,β¹ j

`

˜ _k ÿ

j“2

RFj

ξ_j,k,β¹

j ´ γ

k`1

k

ÿ

j“2

u^p_j,k^β´1

¸

`

˜ γ k`1

k

ÿ

j“2

u^p_j,k^β´1´ γ pβ

¸

`

k

ÿ

j“2

RFyj

Rn,j,β

j

“T_k,n^p1q`T<sub>k,n</sub><sup>p2q</sup>`R^p0q_n `R_n^p1q, (21) with

T_k,n^p1q“

k

ÿ

j“2

´

logRFyj´logRFj

¯ RFj

ξ¹_j,k,β

j `

k

ÿ

j“2

#

´logRFyj

RF_j ´

˜

1´RFyj

RF_j

¸+

RFj

ξ¹_j,k,β j

“T_k,n^p1,1q ` T_k,n^p1,2q. (22)

The termT_k,n^p1,1q is introduced in order to make logarithms of the Kaplan-Meier product appear, leading to manageable sums. Indeed, by definition ofFs_n^KM we find that

logRFy_j“

k

ÿ

i“j

δ<sub>n´i`1,n</sub>log`_i´1

i

˘ and logRF_j “ ´1 γ1

k

ÿ

i“j

ξ_i i `

ˆ

logRF_j` 1 γ1

logZ_n´j`1,n Zn´k,n

˙ . Consequently, defining the following important notations

RF_j,β “RF_j

ˆZ_n´j`1,n Zn´k,n

˙´β

i“2, . . . , k, (23)

and

Si,k,β“ 1 i

i

ÿ

j“2

RFj,β

ξ_j¹

j , i“2, . . . , k, (24)

by inverting sums we obtain T_k,n^p1,1q“

k

ÿ

i“2

„1 γ1

pξi´γq ``

δn´i`1,nilog`_i´1

i

˘` p˘



Si,k,β´

k

ÿ

j“2

ˆ

logRFj` 1 γ1

logZn´j`1,n

Zn´k,n

˙ RFj,β

ξ_j¹ j

“T_k,n^p1,1,1q ´ T_k,n^p1,1,2q. (25)

To summarize,

Tp_kpβq ´ γ₁

1`γ1β “T<sub>k,n</sub><sup>p1,1,1q</sup>´T<sub>k,n</sub><sup>p1,1,2q</sup>`T_k,n^p1,2q`T<sub>k,n</sub><sup>p2q</sup>`R^p0q_n `R^p1q_n . (26) Introducing now the additional notations

c_i “1`ilogi´1

i , A_i,n“ppE_i^pnq´1q ´ pδ_n´i`1,n´pq, and B_i,n“ 1 γ1

b_n,ku^β_i,k^˚γE^pnq_i , and usingp17q, one readily obtains the following formula for the main termT_k,n^p1,1,1q:

T_k,n^p1,1,1q“

k

ÿ

i“2

A_i,nS_i,k,β`

k

ÿ

i“2

B_i,nS_i,k,β`

k

ÿ

i“2

δ<sub>n´i`1,n</sub>c<sub>i</sub>S<sub>i,k,β</sub>` 1 γ1

k

ÿ

i“2

R_n,iS_i,k,β. (27)

In the sequel, we will show that the variablesSi,k,β can be approximated appropriately by _p^γ

β

1

k`1u^p_i,k^β´1. Also, as it is explained in Einmahl et al. (2008), on one hand the parameter p “ γ{γ₁ “ _γ^γ2

1`γ2 is the limit of ppzq “ Ppδ “ 1|Z “ zq as z Ñ 8, and on the other hand the original observations pZ_i, δ_iqiďn

have the same distribution as the variablespZ_i¹, δ¹_iqiďn, wherepZ_i¹qiďnis an independent copy of the sequence pZ_iqiďn, δ_i¹ “IU_iďppZ_i¹qandpU_iqiďndenotes some given i.i.d. sequence of standard uniform random variables (shortened to rv’s), which are independent of the sequencepZ_i¹qiďn. We thus carry on the proof by considering from now on that the observationsδi andZi are related by the formula

δi“IU_iďppZ_iq.

Mimicking what is done in Einmahl et al. (2008), we will later (see proof of Lemma 8) approximate the rv’sδn´i`1,n by i.i.d Bernoulli rv’sIU_iďp.

The main goal will thus be to prove that the termřk

i“2A_i,nS_i,k,β above is (up to a bias term) close to the main random term appearing in Theorem 1

γ pβ

1 k`1

k

ÿ

i“2

tppEi´1q ´ pIU_iďp´pquu^p_i,k^β´1 (28) The other terms inp27qwill be bias or remainder terms, noting that the coefficients ci are close to 0.

The second termT_k,n^p1,1,2q inp26qturns out to be adding to the bias since it only involves the slowly varying functionlF. The treatment of the third termT_k,n^p1,2qabove is very important since it strongly participates to the approximation of a ratio of the formFs_n^KMpxq{Fs_n^KMpyqby the ratioFspxq{Fpyq, for very large values ofs xandy. Such approximation is delicate. Invoking results from survival analysis, we will show however that T_k,n^p1,2qis a remainder term.

Next,T_k,n^p2q is decomposed using the variablespV_j,kqintroduced inp16q:

T_k,n^p2q“ γ k`1

k

ÿ

j“2

´

V_j,k^pβ´u^p_j,k,^β ¯ u^´1_j,k`

k

ÿ

j“2

V_j,k^pβξ_j¹ ´γ j `

k

ÿ

j“2

´

RF_j,β´V_j,k^pβ¯ξ_j¹

j. (29)

According to the definition of ξ¹_j, we can see that the second term of this decomposition is close to

γ k`1

řk

j“2pEj ´1qu^p_j,k^β´1. While this is part of the main term described in p28q, we will find in Proposi- tion 3 that this term is neutralized by another part ofT_k,n^p2q, so thatT_k,n^p2qis just a bias term. FinallyR^p0qn and R^p1qn will also turn out to be remainder terms.

The rest of the section is organised as follows. In subsection 3.1, we set additional notations and state 7

some preliminary approximation results needed in the sequel. In subsection 3.2 we state the asymptotic results for all terms in (26) and conclude the proof.

3.1. Additional notations and important preliminary results

‚First, in the sequel we will regularly work under the following event, for someαą1 arbitrary close to 1, En,α“ @1ďjďk , α^´1uj,kďVj,kďαuj,k

(, (30)

whereuj,kandVj,k are defined inp13qandp16q. According to Shorack and Wellner (1986) (chapter 8), for everyαą1 we have limnÑ8PpEn,αq “1. In the proof section, working ”on the eventEn,α” will thus mean stating bounds or results which are valid with an arbitrary large probability.

‚ Secondly, the remainder term Rn,j defined in the second-order exponential representation of the log- spacingsp17qsatisfy, according to Theorem 2.1 in Beirlant et. al. (2002),

ˇ ˇ ˇ

řk j“i

R_n,j j

ˇ ˇ

ˇ“o_Ppbn,klog_`p_u¹

i,kqq. (31)

‚Thirdly, under assumptionsp1qandp2q, sinceZi“UHpYiq, one can show usingp1qandp4qthat RFj,β“ FspZn´j`1,nq

FspZn´k,nq

ˆZn´j`1,n

Z_n´k,n

˙´β

“V_j,k^pβp1`Cj,k,βq, (32)

whereCj,k,β“Y_n´k,n^´γβ˚DβC^´γβ˚pY˜<sub>k´j`1,k</sub><sup>´γβ˚</sup> ´1qp1`o_Pp1qqandDβ“D´γβD_˚withD“ ´_γ^γ

1D_˚ifβ2ăβ1, D“D1´_γ^γ

1D_˚ ifβ1ďβ2.

‚ Finally, using R´enyi representation (see for example p4.3q in Beirlant et. al. (2004)) and a Taylor ex- pansion, one obtains that for every 2ďjďk,

V_j,k^pβ´u^p_j,k,^β “ ´p_βu^p_j,k,^β

˜ _k ÿ

i“j

E_i^pnq´1 i

¸

´ p_βu^p_j,k,^β

˜ _k ÿ

i“j

1 i ´log

ˆk`1 j

˙¸

` p²_β

2

V˜_j,k^pβplogpV_j,k{u_j,kqq², (33) where ˜Vj,k lies between Vj,k and uj,k. The combination of p32qand p33q thus means that the ratioRFj,β

will be appropriately approximated by the deterministic weightsu^p_j,k,^β . 3.2. Asymptotics for the terms in (26)and conclusion of the proof

The first result stated concerns the termT_k,n^p1,1,1q, which contains the main term of the decomposition of Tp_kpβq ´_1`γ^γ1

1β (see relationsp27qandp28q).

Proposition 1. Under the conditions of Theorem 1, asnÑ 8, we have paq ?

křk

i“2Ai,nSi,k,β“Gn`λbβ`o_Pp1q, where b_β“ ´γ p

p_βp1´pqpDγq_˚β_˚C^´γβ˚{pp_β`γβ_˚q and pDγq_˚ “

$

&

%

γ1D1 if β1ăβ2

´γ2D2 if β2ăβ1

γ1D1´γ2D2 if β1“β2

andGn is equal in distribution to γ pβ

?1 k

k

ÿ

i“2

u^p_i,k^β´1pppE_i´1q ´ pIUiďp´pqq,

where pEiqandpδiqare independent iid samples with distributions standard exponential and standard uniform. The variable Gn is asymptotically centred gaussian distributed with varianceσ²_β“ ^γ

2

p²_β p 2p_β´1. pbq ?

křk

i“2Bi,nSi,k,β“λb_˚`o_Pp1q, whereb_˚“ ´γ²_p^p

βD_˚β_˚C^´γβ˚{ppβ`γβ_˚q.

pcq řk

i“2δ_n´i`1,nc_iS_i,k,β “o_Ppk^´1{2q

pdq řk

i“2R_n,iS_i,k,β“o_Ppk^´1{2q

The following proposition concerns the terms R^p0qn ,R^p1qn ,T_k,n^p1,2qandT_k,n^p1,1,2q. The last two of these terms result from the replacement of the ratios of Kaplan-Meier estimates yRF_j by the ratios of the true survival function valuesRF_j.

Proposition 2. Under the conditions of Theorem 1, asnÑ 8,

paqR^p0qn “opk^´1{2q, pbqR^p1qn “o_Ppk^´1{2q, pcqT_k,n^p1,2q“o_Ppk^´1{2q, pdqT_k,n^p1,1,2q“D₁p1`o_Pp1qqZ_n´k,n^´β1 řk

j“2

ˆ´_Z

n´j`1,n

Zn´k,n

¯´β₁

´1

˙ RF_j,β^ξ

1 j

j ` řk

j“2L_n,jRF_j,β^ξ

1 j

j, where 0ďLn,jďD₁²pZ<sub>n´j`1,n</sub><sup>´β1</sup> ´Z<sub>n´k,n</sub><sup>´β1</sup> q<sup>2</sup>p1`o_Pp1qq.

Moreover,T_k,n^p1,1,2q“bKMpk{nq^γβ˚`o_Ppk^´1{2q, wherebKM is equal to´^γ_p²

βD1β1C^´γβ1{ppβ`γβ1qifβ1ďβ2

and to0 ifβ1ąβ2.

The last result concerns the behaviour ofT_k,n^p2q : it turns out that it only generates a bias term.

Proposition 3. We have

T_k,n^p2q“ ´ pβγ k`1

k

ÿ

j“2

pE_j^pnq´1q

˜ 1 j

j

ÿ

i“2

u^p_i,k^β´1´ 1 p_βu^p_j,k^β´1

¸

´ pβγ k`1

k

ÿ

j“2

pE^pnq_j ´1qu^p_j,k^β´1

˜ _k ÿ

i“j

E_i^pnq´1 i

¸

` bn,k

k`1

k

ÿ

j“2

u^p_j,k<sup>β´1`γβ˚</sup>E<sup>pnq</sup><sub>j</sub> ` bn,k

k`1

k

ÿ

j“2

1

u_j,kpV_j,k^pβ´u^p_j,k,^β qu^γβ_j,k^˚E_j^pnq `

k

ÿ

j“2

V_j,k^pβCj,k,β

ξ_j¹ j

´ pβγ k`1

k

ÿ

j“2

u^p_j,k^β´1

˜ _k ÿ

i“j

1

i ´logk`1 j

¸

E^pnq_j ` ppβq<sup>2</sup>γ 2pk`1q

k

ÿ

j“2

1 u_j,k

V˜_j,k^pβ ˆ

log ˆVj,k

u_j,k

˙˙2

E_j^pnq. (34) Moreover, under the conditions of Theorem 1, when nÑ 8we have

T_k,n^p2q“˜b_˚pk{nq^γβ˚`o_Ppk^´1{2q, where˜b_˚“ ´^γ

2β_˚C^´γβ˚

p_β`γβ<sub>˚</sub> pD<sub>˚</sub>`^D_p^β

βq.

The proofs of all these results can be found in the Appendix. Now, since

?k ˆ

Tp_kpβq ´ γ₁ 1`γ1β

˙

“

?kT_k,n^p1,1,1q`

?kT_k,n^p1,2q´

?kT_k,n^p1,1,2q`

?kT_k,n^p1,1,3q`

?kT_k,n^p2q`

?kR^p0q_n `

?kR^p1q_n and assumption (12) holds, by combination of relation (27) and propositions 1, 2 and 3, we have proved that Theorem 1 holds,i.e. that

? k

ˆ

Tp_kpβq ´ γ₁ 1`γ1β

˙

“G_n`λm<sub>β</sub>`o_Pp1qÝÑ^d Npλm_β, σ_β²q,

because it can be checked thatb_β`b<sub>˚</sub>´b<sub>KM</sub>`˜b_˚is actually equal to the valuem_βdescribed in the statement of Theorem 1.

4. Finite sample comparisons

In this section, we consider a comparison (using finite sample simulations) in terms of observed bias and mean squared error (MSE) of the estimators considered in this paper : pγ^pHq_1,k, pγ_1,k^pWq “γp1,kp0q, pγ1,kpβqwith β ‰0, and pγ_1,k^pBRq. Forγp1,kpβq, we consider three different values ofβ (´1, 0.5 and 1.5). In the expression of pγ^pBRq_1,k , the second order parameter β1 of F should be estimated. Instead, we proceed as in Beirlant et al. (2018) (see equations p13q and p14q therein) by reparametrizing β₁pγ_1,k^pWq by ´ρ₁ and we consider two different values ofρ₁ (´1.5 and´2) in the following formula

pγ_1,k^pBRqpρ1q “γp_1,k^pWq´p1´ρ1q²p1´2ρ1q ρ²₁

˜ Tpk

`´ρ1{pγ^pW_1,k^q˘

´ pγ_1,k^pWq 1´ρ₁

¸ . 9