On two estimates related to the change-point problem

TL;DR

This paper investigates the estimation of a smooth functional of a signal with a discontinuity in Gaussian noise, deriving asymptotic likelihood ratios and comparing Bayesian and maximum likelihood estimators.

Contribution

It introduces new asymptotic likelihood ratio processes for the change-point problem and compares the efficiency of Bayesian and MLE methods.

Findings

Asymptotic likelihood ratio process derived for vanishing noise levels

Bayesian and MLE estimates compared in terms of efficiency

Simulation results illustrate non-asymptotic behavior of estimates

Abstract

We consider the problem of estimating a smooth functional of an unknown signal with discontinuity from Gaussian observations. The signal is a known function that depends on an unknown parameter. This problem is closely related to the famous change-point problem. We obtain an asymptotic likelihood ratio process for the noise level tending to 0. Bayesian and maximum likelihood estimates are constructed and their relative efficiency is studied. Some simulation results and conclusions on non-asymptotic behavior of these estimates are presented.