Asymptotically Optimal Detection of Changes in Stochastic Models with Switching Regimes

Abstract

This paper deals with the problem of asymptotically optimal detection of changes in regime-switching stochastic models. We need to divide the whole obtained sample of data into several sub-samples with observations belonging to different states of a stochastic models with switching regimes. For this purpose, the idea of reduction to a corresponding change-point detection problem is used. Both univariate and multivariate switching models are considered. For the univariate case, we begin with the study of binary mixtures of probabilistic distributions. In theorems 1 and 2 we prove that type 1 and type 2 errors of the proposed method converge to zero exponentially as the sample size tends to infinity. In theorem 3 we prove that the proposed method is asymptotically optimal by the rate of this convergence in the sense that the lower bound in the a priori informational inequality is attained for our method. Several generalizations to the case of multiple univariate mixtures of probabilistic distributions are considered. For the multivariate case, we first study the general problem of classification of the whole array of data into several sub-arrays of observations from different regimes of a multivariate stochastic model with switching states. Then we consider regression models with abnormal observations and switching sets of regression coefficients. Results of a detailed Monte Carlo study of the proposed method for different stochastic models with switching regimes are presented.