Testing Homogeneity in a Mixture Distribution via the L^2 Distance Between Competing Models

• Activity Number: 422
• Type: Contributed
• Date/Time: Wednesday, August 6, 2003 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Richard J. Charnigo*+ and Jiayang Sun
• Companies: Case Western Reserve University and Case Western Reserve University
• Keywords: mixture ; L^2 distance ; data mining

Abstract

Ascertaining the number of components in a mixture distribution is an interesting and challenging problem. We present a new method for testing whether a finite mixture distribution is homogeneous. Our method, the D-test, is based on the L^2 distance between a fitted homogeneous model and a fitted heterogeneous model. For mixture components from standard distributions, our D-test statistic has closed-form expressions in terms of parameter estimates. Thus, our test has potential for data mining applications. The D-test is competitive with the Modified Likelihood Ratio Test of Chen, Chen, and Kalbfleisch (2001) in detecting departures from homogeneity when the mixture components are normal. When the mixture components are exponential, the L^2 separation between the homogeneous and heterogeneous models is less pronounced. Thus, in this case we propose that the measure underlying the L^2 distance be modified according to a suitable weighting function, which is equivalent to transforming the data before applying the D-test. Such a modification produces a generalized D-test that is competitive in this case.