M‐estimation Under a Two‐Sample Semiparametric Model

Abstract

We consider M‐estimation under a two‐sample semiparametric model in which the log ratio of two unknown density functions has a known parametric form. This two‐sample semiparametric model, arising naturally from case‐control studies and logistic discriminant analysis, can be regarded as a biased sampling model. A new class of M‐estimators are constructed on the basis of the maximum semiparametric likelihood estimator of the underlying distribution function. It is shown that the proposed M‐estimators are consistent and asymptotically normally distributed. A simulation study is presented to demonstrate the performance of the proposed M‐estimators.