Monte carlo comparison of wald's, likelihood ratio and rao's tests

This paper, through a simulation study, examines the behaviour of Wald's, the likelihood ratio and Rao's tests for testing (a) simple hypotheses, (b) one-dimensional composite hypotheses, as well as (c) multi-dimensional composite hypotheses. Peers (1971) has shown that none of these three asymptotic test procedures is uniformly superior to the other two tests for testing simple hypotheses versus composite hypotheses. In a series of papers, Chandra and Joshi (1983), Chandra and Mukerjee (1985), and Chandraand Samanta (1988) have claimed that for large sample size, Rao's size-adjusted test is locally more powerful than either the size-adjusted likelihood ratio test or the size-adjusted Wald test, when these tests are performed at a common and sufficiently low level of significance. In the present study, the tests were performed at the standard levels (5% and 1%) of significance, and various local alternatives were considered. For sample size as small as 20, the simulation study appears to support the results of Peers (1971) for testing simple as well as one-dimensional composite hypotheses. However, for testing multi-dimensional composite hypotheses, Rao's size adjusted test was found to be uniformly more powerful than the other two tests. For the cases involving the testing of simple as well as one-dimensional composite hypotheses, simulated results reveal an interesting pattern for the superiority of a test in terms of the space of alternatives, with the likelihood ratio test always occupying the second position. If Rao's test is more powerful in the lower side of the alternative space, then in the upper side Wald's test will be more powerful, and vice versa.