Exact asymptotic distribution of change-point mle for change in the mean of Gaussian sequences

TL;DR

This paper derives exact asymptotic distributions for the change-point maximum likelihood estimator in Gaussian sequences, extending to multivariate cases, and validates the results through simulations and real data application.

Contribution

It provides a novel, exact formula for the asymptotic distribution of the change-point MLE in Gaussian sequences, including multivariate extensions.

Findings

Exact distribution derived for Gaussian sequences with known nuisance parameters.

Distribution extension to multivariate Gaussian processes.

Simulation results confirm accuracy and robustness of the derived distribution.

Abstract

We derive exact computable expressions for the asymptotic distribution of the change-point mle when a change in the mean occurred at an unknown point of a sequence of time-ordered independent Gaussian random variables. The derivation, which assumes that nuisance parameters such as the amount of change and variance are known, is based on ladder heights of Gaussian random walks hitting the half-line. We then show that the exact distribution easily extends to the distribution of the change-point mle when a change occurs in the mean vector of a multivariate Gaussian process. We perform simulations to examine the accuracy of the derived distribution when nuisance parameters have to be estimated as well as robustness of the derived distribution to deviations from Gaussianity. Through simulations, we also compare it with the well-known conditional distribution of the mle, which may be interpreted as a Bayesian solution to the change-point problem. Finally, we apply the derived methodology to monthly averages of water discharges of the Nacetinsky creek, Germany.