Modeling of Extreme Values via Exponential Normalization Compared with Linear and Power Normalization

Haroon Mohamed Barakat,Nourhan Khalil Rakha,Osama Mohareb Khaled

doi:10.3390/sym12111876

Abstract

Several new asymmetric distributions have arisen naturally in the modeling extreme values are uncovered and elucidated. The present paper deals with the extreme value theorem (EVT) under exponential normalization. An estimate of the shape parameter of the asymmetric generalized value distributions that related to this new extension of the EVT is obtained. Moreover, we develop the mathematical modeling of the extreme values by using this new extension of the EVT. We analyze the extreme values by modeling the occurrence of the exceedances over high thresholds. The natural distributions of such exceedances, new four generalized Pareto families of asymmetric distributions under exponential normalization (GPDEs), are described and their properties revealed. There is an evident symmetry between the new obtained GPDEs and those generalized Pareto distributions arisen from EVT under linear and power normalization. Estimates for the extreme value index of the four GPDEs are obtained. In addition, simulation studies are conducted in order to illustrate and validate the theoretical results. Finally, a comparison study between the different extreme models is done throughout real data sets.

Highlights

It has become necessary to study statistical models that have the ability to evaluate these rare phenomena to avoid its dangers due to the sudden rise of some natural harmful phenomena, such as earthquakes, Tsunami, air pollution, and other phenomena
The first stage is to infer the generalized extreme value distributions related to the extreme value theorem (EVT) under exponential normalization
In a spirit of the result of [3,4], we propose applying the peak over threshold (POT) approach based on the EVT under exponential normalization, where we deal with the right tail F ( x ) = 1 − F ( x ), for large x, i.e., we deal with top-order observations

Summary

Introduction

It has become necessary to study statistical models that have the ability to evaluate these rare phenomena to avoid its dangers due to the sudden rise of some natural harmful phenomena, such as earthquakes, Tsunami, air pollution, and other phenomena. In [19], the authors showed that the possible limit laws arisen from (4) attract more DFs than the p-max-stable laws This fact virtually means that the linear and power models may fail to fit the given extreme data, while the exponential model succeeds. The first stage is to infer the generalized extreme value distributions related to the EVT under exponential normalization These asymmetric DFs enable us to apply the BM approach. The second stage is deriving the possible generalized Pareto families of asymmetric distributions relating to the EVT under exponential normalization These families will pave the way to applying the POT approach.

Preliminary Results

BM Approach and GPDEs

Simulation Study

Conclusions