Statistics and Its Interface

Volume 7 (2014)

Number 3

Special Issue on Extreme Theory and Application (Part I)

Guest Editors: Yazhen Wang and Zhengjun Zhang

Point and exact interval estimation for the generalized Pareto distribution with small samples

Pages: 389 – 404

DOI: http://dx.doi.org/10.4310/SII.2014.v7.n3.a9

Authors

Jian He (Department of Statistics and Finance, Business School, Shihezi University, China)

Zhuo Sheng (Department of Mathematical Sciences, Brunel University, London, United Kingdom)

Bing Xing Wang (Department of Mathematics, Zhejiang Gongshang University, China)

Keming Yu (Department of Mathematical Sciences, Brunel University, London, United Kingdom)

Abstract

In extreme value theory, the generalized Pareto distribution (GPD) is used to model another distribution on the tail. Since only a proportion of the data is used, the effective data size for fitting GPD is often small. As statistical properties, especially tail behaviour, of GPD largely depend on its shape parameter, performances of most existing methods are inconsistent when the value of the shape parameter varies. In this paper, we introduce a new method to fit GPD that improves the performance over existing methods for very small samples, in terms of bias and mean square error as well as confidence intervals. The numerical study in this paper also shows that better performance on parameter estimation does not necessarily lead to better performance on quantile estimation.

Keywords

confidence interval, extreme value theory, generalized Pareto distribution, quantile, shape parameter, small sample

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