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A simple iterative procedure is found for the exact null and alternative distributions of likelihood ratio, cumulative sum and related statistics for testing for a change in probability of a sequence of independent binomial random variables. It is concluded that the likelihood ratio test is slightly less powerful than the cumulative sum test near the middle of the sequence, but that the likelihood ratio test is much more powerful near the ends. An example from epidemiology is used to illustrate the results. Under certain conditions the cumulative sum test is identical to a two-sample Kolmogorov-Smirnov test and so the iterative procedure can be used to find the null distribution of Kolmogorov-Smirnov and related statistics.
Keith J. Worsley (Sat,) studied this question.
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