Extreme Value Index Estimation in the Extreme Value Theorem under Non-Linear Normalization

Abstract

The primary goal of this study is to expand the application of the extreme value theorem by developing the modeling of extreme values using non-linear normalization. The issue of estimating the extreme value index (the non-zero extreme value index) under power and exponential normalization is addressed in this study. Under exponential normalization, counterparts of the Hill estimators for the extreme value index estimators under linear normalization are proposed based on the characteristics of the extreme value index, threshold, and the data itself. In addition, based on the generalized Pareto distributions, more condensed and flexible Hill estimators are proposed under power and exponential normalization. These proposed estimators assist us to choose the threshold more flexibly and getting rid of data waste. The R-package runs a thorough simulation analysis to examine the effectiveness of the suggested estimators.

Keywords:

• Extreme value theorem
• generalized extreme value distribution
• generalized pareto distributions
• hill estimators
• maximum likelihood method
• non-linear normalization

Acknowledgements

The authors are immensely grateful to Professor Madhuri S. Mulekar, the Editor in Chief of American Journal of Mathematical and Management Sciences, as well as the anonymous referees for their careful reading of the manuscript and their constructive detailed comments.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Data Availability Statement

The simulated data used to support the findings of this study are included within the article.