Density estimation under a two-sample semiparametric model

We consider estimating a density function under a two-sample semiparametric model in which the log ratio of two density functions is a quadratic function of data. This two-sample semiparametric model, arising naturally from case-control studies and logistic discriminant analysis, can be regarded as a biased sampling model. Under this model, the difference between the two samples is quantified. A kernel-based density estimator is constructed by smoothing the increments of the maximum semiparametric likelihood estimator of the underlying distribution function. The required computation for our method can be accomplished by using the standard statistical software packages for categorical data analysis. We establish some asymptotic results on the proposed kernel density estimator. In addition, we present some results on a simulation study and on the analysis of two data sets to demonstrate the utility of the proposed density estimator.