The Annals of Statistics

Testing uniformity versus a monotone density

Jiayang Sun and Michael Woodroofe

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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.

Article information

Ann. Statist., Volume 27, Number 1 (1999), 338-360.

First available in Project Euclid: 5 April 2002

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Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing 62A10
Secondary: 62G05: Estimation 62E25 62F30: Inference under constraints 60F05: Central limit and other weak theorems 62P20: Applications to economics [See also 91Bxx] 62P99: None of the above, but in this section

Air-conditioning equipments penalized maximum likelihood estimates selection bias nonhomogeneous Poisson process


Woodroofe, Michael; Sun, Jiayang. Testing uniformity versus a monotone density. Ann. Statist. 27 (1999), no. 1, 338--360. doi:10.1214/aos/1018031114.

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