Abstract

Ascertaining the number of components in a mixture distribution is an interesting and challenging problem for statisticians. Chen, Chen, and Kalbeisch (2001) recently proposed a modified likelihood ratio test (MLRT), which is distribution-free and locally most powerful, asymptotically. In this paper we present a new method for testing whether a finite mixture distribution is homogeneous. Our method, the D-test, is based on the L² 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, whereas likelihood ratio-type test statistics do not. Thus, our test has potential for data mining applications. The convergence rate of the D-test statistic under a null hypothesis of homogeneity is established. The D-test is shown to be competitive with the MLRT when the mixture components are normal. The MLRT performs better for small sample sizes when the mixture components are exponential, but in this case there is little visual separation and, hence, little L² separation between the homogeneous and heterogeneous models. Thus, we propose that the measure underlying the L² be modified according to a suitable weight 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 the aforementioned case. After applying our method to a data set in which the observations are measurements of firms' financial performances, we conclude with discussion and remarks.

Citations