- Research Article
- Open Access
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.
- Order Statistic
- Maximum Likelihood Method
- Strong Convergence
- Insurance Policy
- Index Estimator
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. , 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.
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.
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).
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