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Classical penalized likelihood regression problems deal with the case thatthe independent variables data are known exactly. In practice, however, it iscommon to observe data with incomplete covariate information. We are concernedwith a fundamentally important case where some of the observations do notrepresent the exact covariate information, but only a probability distribution.In this case, the maximum penalized likelihood method can be still applied toestimating the regression function. We first show that the maximum penalizedlikelihood estimate exists under a mild condition. In the computation, wepropose a dimension reduction technique to minimize the penalized likelihoodand derive a GACV (Generalized Approximate Cross Validation) to choose thesmoothing parameter. Our methods are extended to handle more complicatedincomplete data problems, such as, covariate measurement error and partiallymissing covariates.

Penalized Likelihood Regression in Reproducing Kernel Hilbert Spaces with Randomized Covariate Data