Extreme Limit Theory of Competing Risks under Power Normalization

TL;DR

This paper develops new probability models for power normalized maxima and minima in competing risks, introducing accelerated max-stable and min-stable distributions, with theoretical properties, estimation methods, and real data applications.

Contribution

It introduces the accelerated p-max/p-min stable distribution family for competing risks, extending classical extreme value theory with new models and inference techniques.

Findings

Efficient approximation of limit scenarios under power normalization

Comparable convergence rates with linear normalization methods

Successful application to real datasets on ozone levels and survival times

Abstract

Advanced science and technology provide a wealth of big data from different sources for extreme value analysis. Classical extreme value theory was extended to obtain an accelerated max-stable distribution family for modelling competing risk-based extreme data in Cao and Zhang (2021). In this paper, we establish probability models for power normalized maxima and minima from competing risks. The limit distributions consist of an extensional new accelerated max-stable and min-stable distribution family (termed as the accelerated p-max/p-min stable distribution), and its left-truncated version. The consistency and asymptotic normality are obtained for the maximum likelihood estimation of the parameters involved in the accelerated p-max and p-min stable distributions when it exists. The limit types of distributions are determined principally by the sample generating process and the interplay among the competing risks, which are illustrated by common examples. Further, the statistical inference concerning the maximum likelihood estimation and model diagnosis of this model was investigated. Numerical studies show first the efficient approximation of all limit scenarios as well as its comparable convergence rate in contrast with those under linear normalization, and then present the maximum likelihood estimation and diagnosis of accelerated p-max/p-min stable models for simulated data sets. Finally, two real datasets concerning annual maximum of ground level ozone and survival times of Stanford heart plant demonstrate the performance of our accelerated p-max and accelerated p-min stable models.