On Limiting Likelihood Ratio Processes of some Change-Point Type Statistical Models

TL;DR

This paper compares two types of limiting likelihood ratio processes in change-point models, showing that under certain conditions, the Poisson-based process can be approximated by the Brownian-based one, aiding statistical inference.

Contribution

It demonstrates the approximation of Poisson likelihood ratios by Brownian ones for small parameters, linking their statistical properties and providing asymptotic insights.

Findings

Poisson likelihood ratios can be approximated by Brownian likelihood ratios for small parameters

The approximation allows for easier computation of variances of estimators

Numerical simulations support the theoretical approximation

Abstract

Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. In this paper we consider two such likelihood ratios. The first one is an exponential functional of a two-sided Poisson process driven by some parameter, while the second one is an exponential functional of a two-sided Brownian motion. We establish that for sufficiently small values of the parameter, the Poisson type likelihood ratio can be approximated by the Brownian type one. As a consequence, several statistically interesting quantities (such as limiting variances of different estimators) related to the first likelihood ratio can also be approximated by those related to the second one. Finally, we discuss the asymptotics of the large values of the parameter and illustrate the results by numerical simulations.