# Strong Convergence Bound of the Pareto Index Estimator under Right Censoring

- Bao Tao
^{1}Email author and - Zuoxiang Peng
^{2}

**2010**:209156

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

© B. Tao and Z. Peng. 2010

**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

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].

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.

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

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.

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

Theorem 2.1.

Theorem 2.2.

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

Theorem 2.3.

## 3. Proofs

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.

Lemma 3.2.

Proof.

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

Lemma 3.3.

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.

Lemma 3.5.

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.

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.

## 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

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