A two-sample nonparametric likelihood ratio test

Abstract

This paper proposes a new test for the hypothesis that two samples have the same distribution. The likelihood ratio test of Portnoy [Portnoy, S. (1988), ‘Asymptotic Behaviour of Likelihood Methods for Exponential Families When the Number of Parameters Tends to Infinity’, Annals of Statistics, 16, 356–366] is applied in the context of the consistent series density estimator of Crain [Crain, B.R. (1974), ‘Estimation of Distributions Using Orthogonal Expansions’, Annals of Statistics, 2, 454–463] and Barron and Sheu [Barron, A.R., and Sheu, C.-H. (1991), ‘Approximation of Density Functions by Sequences of Exponential Families’. Annals of Statistics, 19, 1347–1369]. It is proven that the test, when suitably standardised, is asymptotically standard normal and consistent against any complementary fixed alternative. In comparison with established tests, such as the Kolmogorov–Smirnov, Cramér-von Mises and rank sum, median, and dispersion tests, the proposed tests enjoy broadly comparable finite sample size properties, but vastly superior power properties when considered over a range of different alternatives.

Keywords:

• two sample tests
• series density estimator
• non-parametric likelihood ratio

2000 AMS Subject Classification :

• Primary: 62G10
• Secondary: 62G07

Acknowledgements

Thanks are due to Francesco Bravo, Giovanni Forchini, Les Godfrey, Grant Hillier, Peter Phillips, and the participants at the York Econometrics Workshops, July 2008, to the editor Michael Akritas for suggesting comparisons with rank and median-based tests as well as to two anonymous referees and an associate editor.