On Compound Poisson Processes Arising in Change-Point Type Statistical Models as Limiting Likelihood Ratios

TL;DR

This paper investigates the asymptotic behavior of likelihood ratios in change-point models, showing that compound Poisson process-based likelihood ratios can be approximated by Brownian motion-based ratios for small parameters, with numerical illustrations.

Contribution

It extends previous work by demonstrating that compound Poisson likelihood ratios can be approximated by Brownian motion ratios, broadening the understanding of limiting processes in change-point models.

Findings

Compound Poisson likelihood ratios can be approximated by Brownian motion ratios for small parameters.

The asymptotic behavior of these ratios is analyzed for large parameter values.

Numerical simulations support the theoretical approximations.

Abstract

Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. In a previous paper of one of the authors it was established that one of these likelihood ratios, which is an exponential functional of a two-sided Poisson process driven by some parameter, can be approximated (for sufficiently small values of the parameter) by another one, which is an exponential functional of a two-sided Brownian motion. In this paper we consider yet another likelihood ratio, which is the exponent of a two-sided compound Poisson process driven by some parameter. We establish, that similarly to the Poisson type one, the compound Poisson type likelihood ratio can be approximated by the Brownian type one for sufficiently small values of the parameter. We equally discuss the asymptotics for large values of the parameter and illustrate the results by numerical simulations.