Likelihood inference about a change point in switching autoregression

Abstract

A likelihood approach is considered to the problem of making inferences about the point t = ν in a Gaussian autoregressive sequence {X_t, t = 1 ÷ N} at which the underlying AR(p) parameters undergo a sudden change. The statistics of a loglikelihood function L(n, ν) is investigated over the admissible values n ∈ (p + 1, $\dots$ , N - 1) of a change point ν under validity of hypothesis of a change and no change. The expressions of L(n, ν) implying the loss of plausibility when moving away from the true change point ν are presented, and the probabilities P{ $\bar{v}_{N}$ = ν± r}, r = 0,1,2, $\dots$ , where $\bar{v}_{N}$ is the MLH estimate of a change point ν from the available realization x₁,x₂,…,x_N of {X_t, t = 1 ÷ N} are considered.