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.