On misspecification in cusp-type change-point models

TL;DR

This paper studies the asymptotic behavior of pseudo maximum likelihood estimators in misspecified cusp-type change-point models for inhomogeneous Poisson processes, revealing their convergence properties and limit distributions.

Contribution

It provides new insights into the asymptotic properties of estimators under model misspecification in cusp-type change-point models.

Findings

Estimator converges to the minimizer of Kullback-Leibler divergence.

Normalized estimation error converges to a limit distribution.

Polynomial moments of the error also converge.

Abstract

The problem of parameter estimation by i.i.d. observations of an inhomogeneous Poisson process is considered in situation of misspecification. The model is that of a Poissonian signal observed in presence of a homogeneous Poissonian noise. The intensity function of the process is supposed to have a cusp-type singularity at the change-point (the unknown moment of arrival of the signal), while the supposed (theoretical) and the real (observed) levels of the signal are different. The asymptotic properties of pseudo MLE are described. It is shown that the estimator converges to the value minimizing the Kullback-Leibler divergence, that the normalized error of estimation converges to some limit distribution, and that its polynomial moments also converge.