The distribution of the maximum likelihood estimates of the change point and their relation to random walks

The problem of estimating the change point in a sequence of independent observations is considered. Hinkley (1970) demonstrated that the maximum likelihood estimate of the change point is associated with a two-sided random walk in which the ascending and descending epochs and heights are the key elements for its evaluation. The aim here is to expand the information generated from the random walks and from fluctuation theory and applied to the change point formulation. This permits us to obtain computable expressions for the asymptotic distribution of the change point with respect to convolutions and Laplace transforms of the likelihood ratios. Further, if moment expressions of the likelihood ratios are known, explicit representations of the asymptotic distribution of the change point become accessible up to the second order with respect to the moments. In addition, the rate of convergence between the finite and infinite distribution of the change point distribution is established and it is shown to be of polynomial order.