Penalized likelihood regression in reproducing kernel Hilbert spaces with randomized covariate non-Gaussian data

An Appendix with proofs and tuning details has been added here. Abstract Classical penalized likelihood regression problems deal with the case that the independent variables data are known exactly. In practice, however, it is common to observe data with incomplete covariate information. We are concerned with a fundamentally important case where some of the observations do not represent the exact covariate information, but only a probability distribution. In this case, penalized likelihood method is still applicable to estimate the regression function. We show that penalized likelihood estimate exists under a mild condition. In the computation, we propose a dimension reduction technique to minimize the penalized likelihood and a posterior version of GACV (Generalized Approximate Cross Validation) to choose the smoothing parameter. Our methodology can be extended to handle more complicated cases of incomplete covariate information. For example , covariate measurement error and partially missing covariates can be treated as special cases.