Strong Convergence Bound of the Pareto Index Estimator under Right Censoring

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.