Testing uniformity versus a monotone density

Abstract

The paper is concerned with testing uniformity versus a monotone den-sity. This problem arises in two important contexts, after transformations, testing whether a sample is a simple random sample or a biased sample, and testing whether the intensity function of a nonhomogeneous Poisson process is constant against monotone alternatives. A penalized likelihood ratio test ($P$-test) and a Dip likelihood test ($D$-test) are developed. The $D$-test is analogous to Hartigan and Hartigan’s (1985) Dip test for bump hunting problems. While nonparametric, both the $P$- and $D$-tests are quite efficient in comparison to the most powerful (MP) tests for some simple alternatives and also the Laplace test developed for nonhomogeneous Poisson process. The $P$- and $D$-tests have higher power than the above MP tests under different sets of monotone alternatives and so have greater applicability. Moderate sample size performance and applications of our tests are illustrated via simulations and examination of an air-conditioning equipment data set.