An Empirical likelihood Method in Mixture Models with Incomplete Classifications

In studying the relationship between a binary variable and a covariate, it is very common that the value of the binary variable is missing for some observations, and subsequently make those observations uncategorised. In this paper we show that the uncategorised data can be treated as auxiliary information as in survey sampling literature. We establish a framework of parametric and nonparametric estimation by the empirical likelihood. The proposed empirical likelihood estimators improve the efficiency of estimators based on the categorised samples in the leading order. In a comparative study with the ratio estimator, we reveal robust performance of the empirical likelihood estimators. Possible applications in tax-auditing problem and genetic studies are discussed.