Statistics and Its Interface

Volume 12 (2019)

Number 4

Statistical inference of the generalized Pareto distribution based on upper record values

Pages: 501 – 510



Xu Zhao (College of Applied Science, Beijing University of Technology, Beijing, China)

Xueyan Geng (College of Applied Science, Beijing University of Technology, Beijing, China)

Weihu Cheng (College of Applied Science, Beijing University of Technology, Beijing, China)

Pengyue Zhang (Department of Biomedical Informatics, College of Medicine, The Ohio State University, Columbus, Oh., U.S.A.)


Upper records are important statistics in environmental science and many other fields. Because upper records are crucial for policy making, precise modeling and inference techniques are in high demand. The generalized Pareto distribution (GPD) is commonly adopted by researchers for modeling heavy tail phenomena in many applications. The statistical inference of the GPD upper records is a critical issue in record analysis. Based on upper record data, the current parameter estimation methods of the GPD depend on preassumed shape parameter and only estimate the location and scale parameters. However, the shape parameter is typically unknown in real applications. In this manuscript, we propose a new approach that can estimate all three parameters of the GPD. The proposed estimator is used in conjunction with a moment method and nonlinear weighted least squares theory that minimizes the sum of squared deviations between the upper records and their expectations. In simulation studies, we compare alternative estimators and demonstrate that the new estimator is competitive in terms of the bias and means square error in estimating the shape and scale parameters. In addition, we investigate the performance of different threshold selection procedures by estimating the Value-at-Risk (VaR) of the GPD. Finally, we illustrate the utilization of the proposed methods by analyzing an air pollution data. In this analysis, we provide a detailed guide for selecting the threshold and upper records.


generalized Pareto distribution, extreme values, upper record values, parameter estimation, threshold selection

2010 Mathematics Subject Classification


The authors gratefully acknowledge the support of National Natural Science Foundation of China through Grant No. 11801019, Science and Technology Program of Beijing Education Commission through Grant No. KM201610005020.

Received 17 September 2018

Accepted 19 December 2018

Published 18 July 2019