Likelihood Ratio, Score, arid Wald Statistics in Models with Monotone Functions: Some Comparisons

BANERJEE AND WELLNER (2001) introduced and studied the likelihood ratio statistic for testing the hypothesis that a monotone function takes on a fixed value at a fixed point in the context of estimating the distribution function of the survival time in the interval censoring model. In this paper we continue to use the interval censoring model as a simple "test problem" . We introduce three natural "score statistics" for the same testing problem studied in BANERJEE AND WELLNER (2001) which have natural intepretations in terms of certain (weighted) £2 distances. We compare these new test statistics with an analogue of the classical Wald statistic and the likelihood ratio statistic introduced in BANERJEE AND WELLNER (2001). We first establish limiting distribution theory of the statistics under the null hypothesis, and discuss calculation of the relevant critical points for the test statistics. We then establish the limiting behavior of all five statistics under both fixed and local alternatives. Although the asymptotic theory does allow some qualitative conclusions, it unfortunately does not (yet) lead to explicit quantitative comparison of the power behavior of the five different tests. We therefore also compare the power of five different statistics via a limited Monte-Carlo study. Our preliminary conclusion is that one of the three score tests seem to slightly dominate all the other test statistics, including the likelihood ratio and the Wald statistics. National Science Foundation grant DMS-D07l8I8 National Science Foundation N1A1D 2R01 A1291968-