  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, the maximum penalized likelihood method can be still applied to
estimating the regression function. We first show that the maximum penalized
likelihood estimate exists under a mild condition. In the computation, we
propose a dimension reduction technique to minimize the penalized likelihood
and derive a GACV (Generalized Approximate Cross Validation) to choose the
smoothing parameter. Our methods are extended to handle more complicated
incomplete data problems, such as, covariate measurement error and partially
missing covariates.
