Testing for the number of change points in a sequence of exponential random variables

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Abstract
The literature on change point problems concerns almost exclusively situations with at most one change. Howeverm, in practice it is often not sure that this is so, especially in long observation periods. There is an urgent need for methods to deal with such observations. We give a procedure to determine the number of change points in a sequence of independent exponentially distributed variables, when there is no prior information concerning the (distribution of the) number, location or magnitude of the changes. The procedure is based on partitioning of the likelihood according to a hierarchy of sub–hypotheses. It consists of a sequence of likehood ratio tests of nested hypotheses corresponding to a decreasing number of change points. Here, we consider a maximum of two change points but it is easily generalized for a higher maximum. Since the distribution of the test statistics can, as yet, not be derived analytically, the properties of the procedure were analyzed by Monte Carlo methods. Under the hypotheses of two or less change points the tests appear to be asymptotically independent and also independent of the nuisance parameters. We give critical values for sample sizes up to n=200 and results on the performace of the procedure.