A model for risk assessment of a large earthquake with application to Chilean data

TL;DR

This paper models the waiting time for the next large earthquake using asymptotic distribution theory, proposing a new statistical approach validated through simulations and real Chilean seismic data.

Contribution

It introduces a novel asymptotic exponential model for earthquake waiting times and develops a random cumulative function approach applicable to seismic data analysis.

Findings

Asymptotic exponential distribution accurately models earthquake waiting times.

The proposed random cumulative function satisfies Glivenko-Cantelli and Kolmogorov theorems.

Simulation and real data demonstrate the model's practical utility.

Abstract

We study the asymptotic distribution for the occurrence time of the next large earthquake, by knowing the last large seismic event occurred a long time ago. We prove that, under reasonable conditions, such a distribution is asymptotically exponential with a rate depending on the asymptotic slope of the cumulative intensity function corresponding to a non-homogeneous Poisson process. Moreover, as it is not possible to obtain an empirical cumulative distribution function for the waiting time of the next large earthquake, a random cumulative function based on existing data is stated. We demonstrate that analogous results to the theorems of Glivenko-Cantelli and Kolmogorov are satisfied by this random cumulative function. We conduct a simulation study for detecting in what scenario the approximate distribution of the studied elapsed time performs well. Finally, a real-world data analysis is carried out to illustrate the potential applications of our proposal.