On the asymptotic normality of the extreme value index for right-truncated data

Benchaira, Souad; Meraghni, Djamel; Necir, Abdelhakim

doi:10.1016/j.spl.2015.08.031

Cited by 8 publications

(18 citation statements)

References 19 publications

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“…For brevity, we only prove the third part of the lemma. First, using relation (2) and Lemma 8 (wherein the constants c 0 " 1{q, c 1 " γ 1 {q 2 , c 2 " 2γ 1 2 {q 3 are defined, with q " 1´γ 1 {γ 2 ), it is easily seen that…”

Section: Proof Of Lemmamentioning

confidence: 99%

A Lynden-Bell integral estimator for extremes of randomly truncated data

Worms

2016

Statistics & Probability Letters

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This work deals with the estimation of the extreme value index and extreme quantiles for heavy tailed data, randomly right truncated by another heavy tailed variable. Under mild assumptions and the condition that the truncated variable is less heavy-tailed than the truncating variable, asymptotic normality is proved for both estimators. The proposed estimator of the extreme value index is an adaptation of the Hill estimator, in the natural form of a Lynden-Bell integral. Simulations illustrate the quality of the estimators under a variety of situations.

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Section: Proof Of Lemmamentioning

confidence: 99%

A Lynden-Bell integral estimator for extremes of randomly truncated data

Worms

2016

Statistics & Probability Letters

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“…We shall then highlight a couple of applications of this joint asymptotic normality result, including to the obtention of the asymptotic properties of the tail index estimator introduced by Gardes and Stupfler (2015) when the variable of interest is randomly right-truncated. The question of the convergence of this estimator was considered first by Gardes and Stupfler (2015) under restrictive assumptions, and then by Benchaira et al (2015) using a delicate theoretical argument based on the weighted tail copula process and a joint tail assumption on the observed pair. Our results will make it possible to unify and extend the results of Gardes and Stupfler (2015) and Benchaira et al (2015), without neither resorting to the former's technical conditions nor to the latter's advanced methodology and joint dependence condition.…”

Section: Introductionmentioning

confidence: 99%

“…The question of the convergence of this estimator was considered first by Gardes and Stupfler (2015) under restrictive assumptions, and then by Benchaira et al (2015) using a delicate theoretical argument based on the weighted tail copula process and a joint tail assumption on the observed pair. Our results will make it possible to unify and extend the results of Gardes and Stupfler (2015) and Benchaira et al (2015), without neither resorting to the former's technical conditions nor to the latter's advanced methodology and joint dependence condition. In doing so, we will also be able to give a very simple expression of the asymptotic variance of the limiting normal distribution.…”

Section: Introductionmentioning

confidence: 99%

On a relationship between randomly and non-randomly thresholded empirical average excesses for heavy tails

Stupfler

2019

Extremes

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Motivated by theoretical similarities between the classical Hill estimator of the tail index of a heavy-tailed distribution and one of its pseudo-estimator versions featuring a non-random threshold, we show a novel asymptotic representation of a class of empirical average excesses above a high random threshold, expressed in terms of order statistics, using their counterparts based on a suitable non-random threshold, which are sums of independent and identically distributed random variables. As a consequence, the analysis of the joint convergence of such empirical average excesses essentially boils down to a combination of Lyapunov's central limit theorem and the Cramér-Wold device. We illustrate how this allows to improve upon, as well as produce conceptually simpler proofs of, very recent results about the joint convergence of marginal Hill estimators for a random vector with heavy-tailed marginal distributions. These results are then applied to the proof of a convergence result for a tail index estimator when the heavy-tailed variable of interest is randomly righttruncated. New results on the joint convergence of conditional tail moment estimators of a random vector with heavy-tailed marginal distributions are also obtained.

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“…The sample fraction k = k n being a (random) sequence of integers such that, given n = m = m N , k m → ∞ and k m /m → 0 as N → ∞. Under the tail dependence and the second-order regular variation conditions, Benchaira et al (2015) established the asymptotic normality of this estimator. Recently, Worms and Worms (2016) proposed an asymptotically normal estimator for γ 1 by considering a Lynden-Bell integration with a deterministic threshold.…”

Section: Introductionmentioning

confidence: 99%

Estimating the Second-Order Parameter of Regular Variation and Bias Reduction in Tail Index Estimation Under Random Truncation

Haouas

Necir

Brahimi

2018

J Stat Theory Pract

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In this paper, we propose an estimator of the second-order parameter of randomly righttruncated Pareto-type distributions data and establish its consistency and asymptotic normality. Moreover, we derive an asymptotically unbiased estimator of the tail index and study its asymptotic behaviour. Our considerations are based on a useful Gaussian approximation of the tail product-limit process recently given by Benchaira et al. [Tail product-limit process for truncated data with application to extreme value index estimation. Extremes, 2016; 19: 219-251] and the results of Gomes et al. [Semi-parametric estimation of the second order parameter in statistics of extremes. Extremes, 2002; 5: 387-414]. We show, by simulation, that the proposed estimators behave well, in terms of bias and mean square error.

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On the asymptotic normality of the extreme value index for right-truncated data

Cited by 8 publications

References 19 publications

A Lynden-Bell integral estimator for extremes of randomly truncated data

A Lynden-Bell integral estimator for extremes of randomly truncated data

On a relationship between randomly and non-randomly thresholded empirical average excesses for heavy tails

Estimating the Second-Order Parameter of Regular Variation and Bias Reduction in Tail Index Estimation Under Random Truncation

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