Journal of Probability and Statistics
Volume 2012 (2012), Article ID 537474, 15 pages
http://dx.doi.org/10.1155/2012/537474
Research Article

Testing Homogeneity in a Semiparametric Two-Sample Problem

Yukun Liu,1 Pengfei Li,2 and Yuejiao Fu3

1Department of Statistics and Actuarial Science, School of Finance and Statistics, East China Normal University, Shanghai 200241, China
2Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, Canada N2L 3G1
3Department of Mathematics and Statistics, York University, Toronto, ON, Canada M3J 1P3

Received 18 November 2011; Accepted 24 January 2012

Academic Editor: Yongzhao Shao

Copyright © 2012 Yukun Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study a two-sample homogeneity testing problem, in which one sample comes from a population with density 𝑓 ( 𝑥 ) and the other is from a mixture population with mixture density ( 1 − 𝜆 ) 𝑓 ( 𝑥 ) + 𝜆 𝑔 ( 𝑥 ) . This problem arises naturally from many statistical applications such as test for partial differential gene expression in microarray study or genetic studies for gene mutation. Under the semiparametric assumption 𝑔 ( 𝑥 ) = 𝑓 ( 𝑥 ) 𝑒 𝛼 + 𝛽 𝑥 , a penalized empirical likelihood ratio test could be constructed, but its implementation is hindered by the fact that there is neither feasible algorithm for computing the test statistic nor available research results on its theoretical properties. To circumvent these difficulties, we propose an EM test based on the penalized empirical likelihood. We prove that the EM test has a simple chi-square limiting distribution, and we also demonstrate its competitive testing performances by simulations. A real-data example is used to illustrate the proposed methodology.