Multiple change‐point models for time series

Abstract

The “Bayes‐type” method of deriving change‐point test statistics was introduced by Chernoff and Zacks (1964). Other authors subsequently adapted this approach and derived Bayes‐type statistics for at most one change (AMOC), and for multiple change points, under a variety of model formulations. Asymptotic distribution theory has always been limited to the AMOC statistics because of the perceived complexity of multiple change‐point statistics. In this article, it is shown that the Bayes‐type statistic derived to test for multiple change points is directly proportional to the AMOC statistic. This result immediately provides distributional results for Bayes‐type multiple change‐point statistics. In addition, it fundamentally alters the current understanding of the AMOC statistic. It follows from this result that the Bayes‐type statistic derived under AMOC conditions in fact tests for at least one change (ALOC), even though the statistic is derived under AMOC formulation. Under asymptotic consideration, the result also extends to the case of model errors following a stationary process. As an example, the classical Nile River data are revisited and analyzed for the presence of multiple change points.