Projection density estimation under a mm-sample semiparametric model

An mm-sample semiparametric model in which the ratio of m-1m-1 probability density functions with respect to the mth is of a known parametric form without reference to any parametric model is considered. This model arises naturally from retrospective studies and multinomial logistic regression model. A projection density estimator is constructed by smoothing the increments of the maximum semiparametric empirical likelihood estimator of the underlying distribution function, using the combined data from all the samples. Some asymptotic results on the proposed projection density estimator are established. Connections between our estimator and kernel semiparametric density estimator are pointed out. Some results from simulations and from the analysis of two real data sets are presented.