Strong Convergence Bound of the Pareto Index Estimator under Right Censoring
© B. Tao and Z. Peng. 2010
Received: 25 October 2009
Accepted: 11 April 2010
Published: 18 May 2010
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.
Estimating the Pareto index is very important in theoretical analysis and practical applications of extreme value theory; for example, Embrechts et al. , Reiss and Thomas  andreferences therein. For recent work on estimating extreme value index, see Beirlant and Guillou , Fraga Alves [4, 5], Gomes et al. , Gomes and Henriques Rodrigues , and Li et al. [8, 9].
Mason  proved weak consistency of for any sequence , and Deheuvels et al.  proved its strong consistency for any sequence with The strong convergence bounds of the Hill estimator have been considered by Peng and Nadarajah  under second-order regularly varying conditions.
as the estimator of , where . This estimator reduces to the Hill estimator in the absence of censoring. Beirlant and Guillou  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
By using Theorems 2.1 and 2.2, we can deduce the following theorem easily.
The following auxiliary results are necessary for the proofs of the main results. The first two results are correct due to Wellner .
Notice that is uniformly distributed on (0, 1); the result follows from Wellner .
The result follows from Deheuvels and Mason .
Based on the above lemmas, we prove Theorems 2.1 and 2.2.
Proof of Theorem 2.1.
Combining (3.12), (3.13) with (3.14), we complete the proof.
Proof of Theorem 2.2.
Combining (3.18), (3.19) with (3.20), we complete the proof.
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).
- 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
- 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
- Beirlant J, Guillou A: Pareto index estimation under moderate right censoring. Scandinavian Actuarial Journal 2001, 101(2):111–125.MathSciNetView ArticleMATHGoogle Scholar
- Fraga Alves MI: A location invariant Hill-type estimator. Extremes 2001, 4(3):199–217. 10.1023/A:1015226104400MathSciNetView ArticleMATHGoogle Scholar
- 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
- 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
- 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
- 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
- Li J, Peng Z, Nadarajah S: Asymptotic normality of location invariant heavy tail index estimator. to appear in Extremes to appear in ExtremesGoogle Scholar
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Drees H: On smooth statistical tail functionals. Scandinavian Journal of Statistics 1998, 25(1):187–210. 10.1111/1467-9469.00097MathSciNetView ArticleMATHGoogle Scholar
- 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
- 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
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