A Lynden-Bell integral estimator for the tail index of right-truncated data with a random threshold

Haouas, Nawel; Necir, Abdelhakim; Meraghni, Djamel; Brahimi, Brahim

doi:10.16929/as/2017.1159.97

Cited by 3 publications

(4 citation statements)

References 12 publications

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“…same behavior which actually is noticed before byHaouas et al (2018). The optimal sample fractions k * of each tail index estimator are given in Tables 1…”

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confidence: 74%

Semiparametric tail-index estimation for randomly right-truncated heavy-tailed data

Mancer¹,

Necir²,

Benchaira³

2021

Preprint

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It was shown that when one disposes of a parametric information of the truncation distribution, the semiparametric estimator of the distribution function for truncated data (Wang, 1989) is more efficient than the nonparametric one. On the basis of this estimation method, we derive an estimator for the tail index of Paretotype distributions that are randomly right-truncated and establish its consistency and asymptotic normality. The finite sample behavior of the proposed estimator is carried out by simulation study. We point out that, in terms of both bias and root of the mean squared error, our estimator performs better than those based on nonparametric estimation methods. An application to a real dataset of induction times of AIDS diseases is given as well.

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“…same behavior which actually is noticed before byHaouas et al (2018). The optimal sample fractions k * of each tail index estimator are given in Tables 1…”

supporting

confidence: 74%

Semiparametric tail-index estimation for randomly right-truncated heavy-tailed data

Mancer¹,

Necir²,

Benchaira³

2021

Preprint

Self Cite

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“…We point out that the two estimators trueγ^1)(boldBboldMboldN and trueγ^1)(boldW have almost the same behavior which actually was noticed before by Ref. [15]. The optimal sample fractions and estimate values of the tail index obtained through the three estimators are given in Tables 1–4.…”

Section: Simulation Studymentioning

confidence: 99%

“…In a simulation study, [15] compared this estimator with trueγ^1)(boldBboldMboldN. They pointed out that both estimators have similar behaviors in terms of biases and mean squared errors.…”

Section: Introductionmentioning

confidence: 99%

Semiparametric tail-index estimation for randomly right-truncated heavy-tailed data

Mancer

Necir

Benchaira

2022

AJMS

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PurposeThe purpose of this paper is to propose a semiparametric estimator for the tail index of Pareto-type random truncated data that improves the existing ones in terms of mean square error. Moreover, we establish its consistency and asymptotic normality.Design/methodology/approachTo construct a root mean squared error (RMSE)-reduced estimator of the tail index, the authors used the semiparametric estimator of the underlying distribution function given by Wang (1989). This allows us to define the corresponding tail process and provide a weak approximation to this one. By means of a functional representation of the given estimator of the tail index and by using this weak approximation, the authors establish the asymptotic normality of the aforementioned RMSE-reduced estimator.FindingsIn basis on a semiparametric estimator of the underlying distribution function, the authors proposed a new estimation method to the tail index of Pareto-type distributions for randomly right-truncated data. Compared with the existing ones, this estimator behaves well both in terms of bias and RMSE. A useful weak approximation of the corresponding tail empirical process allowed us to establish both the consistency and asymptotic normality of the proposed estimator.Originality/valueA new tail semiparametric (empirical) process for truncated data is introduced, a new estimator for the tail index of Pareto-type truncated data is introduced and asymptotic normality of the proposed estimator is established.

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“…This problem has only been considered very recently, starting with Gardes and Stupfler (2015), who were ultimately interested in the estimation of extreme quantiles of Y . Several studies have since then proposed alternative techniques for tail index estimation in this context; we refer to Benchaira et al (2016a, b), Worms and Worms (2016) and Haouas et al (2017). The random right-truncation context should not be mistaken for random right-censoring, where the available information is made of the pairs (min(Y i , T i ), 1 {Y i ≤T i } ), 1 ≤ i ≤ n. The latter context has received a substantial amount of attention over the last decade: we refer to Beirlant et al (2007Beirlant et al ( , 2010Beirlant et al ( , 2016, Einmahl et al (2008), Gomes andNeves (2011), Ndao et al (2014), Sayah et al (2014), Worms and Worms (2014), Brahimi et al (2015), Ndao et al (2016), Stupfler (2016), Dierckx et al (2018) and Stupfler (2019).…”

Section: Convergence Of a Tail Index Estimator For Right-truncated Samentioning

confidence: 99%

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

Stupfler

2019

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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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A Lynden-Bell integral estimator for the tail index of right-truncated data with a random threshold

Cited by 3 publications

References 12 publications

Semiparametric tail-index estimation for randomly right-truncated heavy-tailed data

Semiparametric tail-index estimation for randomly right-truncated heavy-tailed data

Semiparametric tail-index estimation for randomly right-truncated heavy-tailed data

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

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