Estimation and Testing Procedures in General Contamination Models
by
Hongying Dai
Department of Statistics, University of Kentucky
Coauthors: Richard Charnigo (Department of Statistics, University of Kentucky)

Lack of identifiability of parameters and estimation of parameters on a boundary make inference in mixture models challenging to statisticians. In homogeneity tests, the maximum likelihood estimators are neither consistent nor asymptotically normal; thus the standard results for likelihood ratio tests cannot be applied in this situation. The contamination model, where a known density is perturbed by a density with an unknown parameter vector, can fit the data well when a sample from one population is contaminated by a sample from another population. The contamination models in this work cover a wide class of parametric families. We address nonidentifiability issues by using a reparametrization approach similar to Lemdani and Pons (1999). The modified maximum likelihood estimators (MMLEs), first proposed by Chen, Chen and Kalbfleisch (2001) in a different setting, are applied for estimation of models under reparametriztion. In this work, we show that the large sample behaviors of the MMLEs follow simple analytical forms. In particular, the MMLE of the mixing parameter under reparametrization is asymptotically normal.

We develop both the D-test and the modified likelihood ratio test for homogeneity in contamination models with vector parameters. The D-test with a weighting function, originally proposed by Charnigo and Sun (2004), is computationally effective and competitive with the likelihood ratio type tests. The limiting null distribution of these two test statistics is ^2_k distribution regardless of the parametric family, making the two tests robust and easy to be applied in practice. The performances of these two tests are assessed by Monte-Carlo simulations.

Keywords: contamination model, mixture, D-test, modified likelihood ratio test

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Date received: October 12, 2005