Merging information for a semiparametric projection density estimation

A semiparametric density estimator is proposed under a m-sample density ratio model, which specifies that the ratio of m−1 probability density functions with respect to the mth is of a known parametric form without reference to any parametric model. 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 likelihood estimator of the underlying distribution function, using the combined data from all the samples. We also establish some asymptotic results on the proposed projection density estimator.