M-estimation under a two-sample semiparametric model

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.