Tests against tree order restriction in Poisson intensities

Abstract

We consider a situation in which one wishes to compare several Poisson intensities with a control or standard when it is believed that the intensities are higher than the control. For instance, if λ1 is the average accident rate per k.m. of a truck driver who has undergone an extensive training in driving and if λj, for j=2,3,...,k, are the average accident rates of the jth truck drivers without any prior training; and if the intensities are believed to produce at least as large an intensity as the control, then one would expect that λ1 ≤ λj for j=2,3,…,k. If xi is the number of accidents incurred by the ith of the k truck drivers and ti be the number of k.m. he drove and λi be the average accident rate per k.m., we can mathematically formulate this situation as follows: Suppose X1,…,Xk are independent Poisson variables with means μi = λiti and let Ho: λ1= λ2=…= λk and H1: λ1 ≤ λj , for j=2,3,…,k, where λ1 is the control intensity and λj, for j =2,3,...,k, are the other intensities. The ordering specified by H1 is called a tree ordering. We are interested in testing Ho versus H1-Ho . The likelihood ratio test for Ho versus H1- Ho is computed and we derive the asymptotic distribution of it. Robertson and Wegman (1978) considered order restricted tests for members of the exponential family. Their results can be applied in the testing situation considered here only if the ti are all equal. Some results are also obtained in the literature for other order restrictions (Magel & Wright (1984) and Barmi et al.(1996)). In this study we obtain explicit formulae for the null distribution of the test statistic under tree ordering.