Change - Point Estimation Under Asymmetric

In the asymptotic setting of the change-point estimation problem the limiting behavior of Bayes procedures for a general loss function is studied. It is demonstrated that the distribution of the diierence between the Bayes estimator and the parameter converges to the distribution of a (nondegenerate) random variable. The sequence of minimum Bayes risks is shown to converge to its supremum, and an explicit formula for this limit is obtained for the geometric prior distribution. Special loss functions, in particular, the linex loss, are considered in detail. 1. Introduction and Summary. In this paper we study the classical change-point estimation problem in the Bayesian setting. Point estimation of the change moment, when the distribution of the sample switches from one distribution to another, was initiated in 6] where the limiting distribution of the maximum likelihood estimator was derived. The convergence results for various non-parametric statistics are obtained in 4], 5]. More references in the area of the non-parametric change-point analysis can be found in the monograph 2]. In this paper the point estimation of the change-point parameter is considered after the data has been observed and the change-point is known to occur. This situation arises in the quality control problems when the last produced items are defective or when the production process gets out of control, in epidemiological studies when the epidemic has started during the given time period, etc. The assumption of the paper about the known form of the pre-and after-change distributions is not very realistic, but is useful to determine a \yardstick" for the performance of the rules constructed without knowledge of these distributions.