Abstract
SUMMARY Fu & Curnow (1990) derive recursive equations to find the level of significance and power of a likelihood ratio test for a changed segment of specified length, based on the scan statistic, the maximum number of successes within the specified length. Their method is computationally feasible for segment lengths of 20 or less. We present and evaluate highly accurate approximations as well as bounds for the power function of this test that are computationally feasible even for very large segment lengths. We also evaluate power when the duration of the increased length used in the test statistic does not correspond to the actual length. Scientists dealing with data sequences modelled by independent Bernoulli trials frequently seek criteria to detect changes in the underlying process. Certain criteria have been developed to be sensitive to situations in which the probability of success on an individual trial changes at a point in the sequence. Fu & Curnow (1990) consider a version of a two change-point problem with known distance, m, between the change points, and with the trial probability of success at the two ends being equal. They give recurrence relations for the probability of type I error and the power. Computational difficulties arise on even large computers when the number of trials is large. In the present paper, we derive highly accurate approximations that allow computation of power for all segment lengths. Consider a sequence of N independent 0/1 random variables X1, X2,.. ., XN. Let '1' denote success, with pr (X, = 1) = i(t). We wish to test the null hypothesis of a constant probability of success HO: i(t) = i (t = 1, . . . , N) against the alternative that some contiguous sequence of m events starting at an unknown trial z has a higher probability of success. Fu & Curnow (1990) note that the generalized likelihood ratio test for this alternative rejects Ho for large values of the scan statistic
Published Version
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