Open Access

Strong Convergence Bound of the Pareto Index Estimator under Right Censoring

Journal of Inequalities and Applications20102010:209156

https://doi.org/10.1155/2010/209156

Received: 25 October 2009

Accepted: 11 April 2010

Published: 18 May 2010

Abstract

Let be a sequence of positive independent and identically distributed random variables with common Pareto-type distribution function as , where represents a slowly varying function at infinity. In this note we study the strong convergence bound of a kind of right censored Pareto index estimator under second-order regularly varying conditions.

1. Introduction

A distribution is said to be of Pareto-type if there exists a positive constant such that
(1.1)
where is a slowly varying function at infinity, that is,
(1.2)

The parameter is called the Pareto index.

Estimating the Pareto index is very important in theoretical analysis and practical applications of extreme value theory; for example, Embrechts et al. [1], Reiss and Thomas [2] andreferences therein. For recent work on estimating extreme value index, see Beirlant and Guillou [3], Fraga Alves [4, 5], Gomes et al. [6], Gomes and Henriques Rodrigues [7], and Li et al. [8, 9].

Suppose that is a sequence of positive independent and identically distributed (i.i.d.) random variables with common distribution function , and let denote the order statistics of . By using maximum likelihood method, Hill [10] introduced the following well-known estimator of , that is,
(1.3)

Mason [11] proved weak consistency of for any sequence , and Deheuvels et al. [12] proved its strong consistency for any sequence with The strong convergence bounds of the Hill estimator have been considered by Peng and Nadarajah [13] under second-order regularly varying conditions.

Recently Beirlant and Guillou [3] proposed a new kind of Hill estimator in case of the sample being right censored. In actuarial setting, most insurance policies limit the claim size and reinsure the large claim size exceeding the given level. So the observed claim size series are right censored. Suppose that are the first ascending order statistics of where is an integer random variable. Define
(1.4)

as the estimator of , where . This estimator reduces to the Hill estimator in the absence of censoring. Beirlant and Guillou [3] proved weak and strong consistency, and asymptotic normality of . In this paper, we consider the strong convergence bound of this Pareto index estimator under second-order regularly varying conditions.

2. Main Results

Denote ; then (1.1) is equivalent to
(2.1)

In order to investigate the strong convergence bound of , we require knowing the convergence rate of (2.1). For this reason, we need the following second-order regular condition.

Suppose that there exists a measurable function satisfying , and a function such that for all ,
(2.2)
Then must be of the form for some ( as ), and is the regularly varying index of , that is, ; for example, de Haan and Stadtmüller [14]. (2.2) holds if and only if for all ,
(2.3)

In order to obtain the strong convergence bound of , we give the following two results firstly.

Theorem 2.1.

Suppose that (2.2) holds, and further assume that , for some , and as , where Then
(2.4)

Theorem 2.2.

Suppose that (2.2) holds, and further assume that , , and as , where Then
(2.5)

By using Theorems 2.1 and 2.2, we can deduce the following theorem easily.

Theorem 2.3.

Suppose that (2.2) holds and assume that ,     , for some , , and as where ; then
(2.6)

3. Proofs

Suppose that are independent and identically distributed random variables with common distribution function . Let denote the order statistics of . It is easy to see that . For the sake of simplicity, define as
(3.1)

The following auxiliary results are necessary for the proofs of the main results. The first two results are correct due to Wellner [15].

Lemma 3.1.

If and , then
(3.2)

Lemma 3.2.

Suppose that , , and . Then
(3.3)

Proof.

Applying Lemma 3.1, we find
(3.4)

Notice that is uniformly distributed on (0, 1); the result follows from Wellner [15].

Lemma 3.3.

Let , and . Then
(3.5)

Proof.

The result follows from Deheuvels and Mason [16].

The following bound of (2.3) is from Drees [17]; for example, Theorem of de Haan and Ferreira [18].

Lemma 3.4.

If (2.2) holds, then for every , there exists such that for and ,
(3.6)

Lemma 3.5.

Assume that , for some and as where Suppose are order statistics from parent with distribution function for some . Then
(3.7)

Proof.

The proof is similar to the proof of Lemma (i) in Dekkers et al. [19]; for example, Lemma of Beirlant and Guillou [3].

Based on the above lemmas, we prove Theorems 2.1 and 2.2.

Proof of Theorem 2.1.

We only prove the case for . For the case for , the proof is similar. Clearly,
(3.8)
where
(3.9)
for . By using Lemmas 3.1 and 3.4, for sufficiently large , we have
(3.10)
Noting that for and by using Lemma 3.5 we find
(3.11)
Since and , by (3.10), and (3.11), and Lemma 3.1, it follows that
(3.12)
by letting . Similarly,
(3.13)
From Lemmas 3.2, and 3.3, and the conditions provided by Theorem 2.1, we have
(3.14)

Combining (3.12), (3.13) with (3.14), we complete the proof.

Proof of Theorem 2.2.

We only prove the case of . Clearly,
(3.15)
where
(3.16)
By using Lemmas 3.1 and 3.4, for sufficiently large , we have
(3.17)
Since , by using Lemma 3.1 and the conditions provided by this theorem, we have
(3.18)
Similarly,
(3.19)
By using Lemma 3.2 and the conditions of Theorem 2.2, it follows that
(3.20)

Combining (3.18), (3.19) with (3.20), we complete the proof.

Declarations

Acknowledgments

The authors would like to sincerely thank the Editor-in-Chief and the anonymous referees for their valuable comments and suggestions on a previous draft, which resulted in the present version of the paper. This paper was partially supported by Chongqing Municipal Education Commission Projects (No. KJ100726, KJ080725) and Chongqing NSF (No. 2009BB8221).

Authors’ Affiliations

(1)
College of Mathematics and Statistics, Chongqing Technology and Business University
(2)
School of Mathematics and Statistics, Southwest University

References

  1. Embrechts P, Klüppelberg C, Mikosch T: Modelling Extremal Events: For Insurance and Finance, Applications of Mathematics. Volume 33. Springer, Berlin, Germany; 1997:xvi+645.View ArticleMATHGoogle Scholar
  2. Reiss R-D, Thomas M: Statistical Analysis of Extreme Values: From Insurance, Finance, Hydrology and Other Fields. Birkhäuser, Basel, Switzerland; 1997:xviii+316.View ArticleMATHGoogle Scholar
  3. Beirlant J, Guillou A: Pareto index estimation under moderate right censoring. Scandinavian Actuarial Journal 2001, 101(2):111–125.MathSciNetView ArticleMATHGoogle Scholar
  4. Fraga Alves MI: A location invariant Hill-type estimator. Extremes 2001, 4(3):199–217. 10.1023/A:1015226104400MathSciNetView ArticleMATHGoogle Scholar
  5. Fraga Alves MI, Gomes MI, de Haan L, Neves C: Mixed moment estimator and location invariant alternatives. Extremes 2009, 12(2):149–185. 10.1007/s10687-008-0073-3MathSciNetView ArticleMATHGoogle Scholar
  6. Gomes MI, de Haan L, Rodrigues LH: Tail index estimation for heavy-tailed models: accommodation of bias in weighted log-excesses. Journal of the Royal Statistical Society. Series B 2008, 70(1):31–52.MathSciNetMATHGoogle Scholar
  7. Gomes MI, Henriques Rodrigues L: Tail index estimation for heavy tails: accommodation of bias in the excesses over a high threshold. Extremes 2008, 11(3):303–328. 10.1007/s10687-008-0059-1MathSciNetView ArticleMATHGoogle Scholar
  8. Li J, Peng Z, Nadarajah S: A class of unbiased location invariant Hill-type estimators for heavy tailed distributions. Electronic Journal of Statistics 2008, 2: 829–847. 10.1214/08-EJS276MathSciNetView ArticleMATHGoogle Scholar
  9. Li J, Peng Z, Nadarajah S: Asymptotic normality of location invariant heavy tail index estimator. to appear in Extremes to appear in ExtremesGoogle Scholar
  10. Hill BM: A simple general approach to inference about the tail of a distribution. The Annals of Statistics 1975, 3(5):1163–1174. 10.1214/aos/1176343247MathSciNetView ArticleMATHGoogle Scholar
  11. Mason DM: Laws of large numbers for sums of extreme values. The Annals of Probability 1982, 10(3):754–764. 10.1214/aop/1176993783MathSciNetView ArticleMATHGoogle Scholar
  12. Deheuvels P, Haeusler E, Mason DM: Almost sure convergence of the Hill estimator. Mathematical Proceedings of the Cambridge Philosophical Society 1988, 104(2):371–381. 10.1017/S0305004100065531MathSciNetView ArticleMATHGoogle Scholar
  13. Peng Z, Nadarajah S: Strong convergence bounds of the Hill-type estimator under second-order regularly varying conditions. Journal of Inequalities and Applications 2006, 2006:-7.Google Scholar
  14. de Haan L, Stadtmüller U: Generalized regular variation of second order. Journal of the Australian Mathematical Society. Series A 1996, 61(3):381–395. 10.1017/S144678870000046XView ArticleMathSciNetMATHGoogle Scholar
  15. Wellner JA: Limit theorems for the ratio of the empirical distribution function to the true distribution function. Probability Theory and Related Fields 1978, 45(1):73–88.MathSciNetMATHGoogle Scholar
  16. Deheuvels P, Mason DM: The asymptotic behavior of sums of exponential extreme values. Bulletin des Sciences Mathématiques. 2e Série 1988, 112(2):211–233.MathSciNetMATHGoogle Scholar
  17. Drees H: On smooth statistical tail functionals. Scandinavian Journal of Statistics 1998, 25(1):187–210. 10.1111/1467-9469.00097MathSciNetView ArticleMATHGoogle Scholar
  18. de Haan L, Ferreira A: Extreme Value Theory: An Introduction, Springer Series in Operations Research and Financial Engineering. Springer, New York, NY, USA; 2006:xviii+417.View ArticleGoogle Scholar
  19. Dekkers ALM, Einmahl JHJ, de Haan L: A moment estimator for the index of an extreme-value distribution. The Annals of Statistics 1989, 17(4):1833–1855. 10.1214/aos/1176347397MathSciNetView ArticleMATHGoogle Scholar

Copyright

© B. Tao and Z. Peng. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.