Estimation of Cusp Location of Stochastic Processes: a Survey

TL;DR

This survey reviews recent advances in estimating the cusp location in various stochastic models, highlighting the properties, efficiency, and asymptotic behavior of estimators like MLE and Bayes, with numerical validation.

Contribution

It compiles and discusses recent results on cusp location estimation across multiple stochastic models, emphasizing asymptotic properties and efficiency of estimators.

Findings

MLE and Bayes estimators exhibit specific asymptotic behaviors.

Log-likelihood ratios converge to fractional Brownian motion.

Numerical simulations support theoretical results.

Abstract

We present a review of some recent results on estimation of location parameter for several models of observations with cusp-type singularity at the change point. We suppose that the cusp-type models fit better to the real phenomena described usually by change point models. The list of models includes Gaussian, inhomogeneous Poisson, ergodic diffusion processes, time series and the classical case of i.i.d. observations. We describe the properties of the maximum likelihood and Bayes estimators under some asymptotic assumptions. The asymptotic efficiency of estimators are discussed as well and the results of some numerical simulations are presented. We provide some heuristic arguments which demonstrate the convergence of log-likelihood ratios in the models under consideration to the fractional Brownian motion.