Tests for and against trends among Poisson intensities
Rhonda C. Magel, Farroll T. Wright · Lecture notes-monograph series · 1984
Suppose one observes independent Poisson processes with unknown intensities λ,, i = 1, ... , k, and that apriori it is believed that these intensities satisfy a known ordering.For preliminary analysis, it might be desirable to test for homogeneity among the intensities and, of course, one would want a test that utilizes the information in the ordering.Let ί, denote the length of time for which the ith process was observed.The case in which the ί, are equal has been studied in the literature.We develop the conditional likelihood ratio test for arbitrary /,.This test is equivalent to the unconditional likelihood ratio test, but leads to an interesting multinomial testing situation, ie.testing for homogeneity of /?,//, versus a trend among the /?,//,, where the pj are the cell probabilities.If the number of trials in the multinomial setting, or the total number of occurrences in the Poisson processes, is large, then the test statistic has an approximate chi-bar-squared distribution which has been studied in the literature.Results of a Monte Carlo study comparing this test with the maximin test developed by Lee (1980) are discussed.Similar results are also obtained for testing the null hypothesis that the intensities satisfy the prescribed ordering.1. Introduction.Barlow, Bartholomew, Bremner and Brunk (1972) discuss the problem of estimating a finite sequence of Poisson intensities which are assumed to be nonincreasing.For instance, consider a system which is observed for t λ units of time with X λ failures, is then modified in an attempt to improve its performance, is observed for t 2 units of time with X 2 failures, is modified again, and this is repeated until it is observed for the kth time for t k units of time with X k failures.If it is believed that the modifications will not harm the system's performance, then one might wish to estimate the vector of intensities, λ = (λ 1? ... , λ k ), subject to λ\ ^ ... ^ \ k .It would also be of interest to test for homogeneity among the intensities with the alternative λι 2* ... ^ λ k and λ x > λ k , or if the assumption concerning the modification were in question, one could test λj ^ ... ^ k k against λ, λ, for some i«j).The hypothesis H λ stipulates that λ = {\ λ , ... , \ k ) is isotonic (with respect to «) and we suppose that λ (0) is isotonic.We consider the likelihood ratio test (lit) for HQ versus //, -H o and H λ versus H 2 conditional on Σ* =1 X, = n.While it will be shown that the conditional test is equivalent to the unconditional lit, it does lead to an interesting multinomial testing situation.We also know that for k = 2 it is UMP unbiased.(See Ferguson (1967, p. 228)).Robertson and Wegman (1978) consider order restricted tests for members of the exponential family, but their work requires that the sample sizes be equal.Their results can be applied in the testing situation considered here only if the t t are all equal.Bos well (1966)