Nonparametric estimation of multivariate CDF with categorical and continuous data

In this paper we construct a nonparametric kernel estimator to estimate the joint multivariate cumulative distribution function (CDF) of mixed discrete and continuous variables. We use a data-driven cross-validation method to choose optimal smoothing parameters which asymptotically minimize the mean integrated squared error (MISE). The asymptotic theory of the proposed estimator is derived, and the validity of the cross-validation method is proved. We provide sufficient and necessary conditions for the uniqueness of optimal smoothing parameters when the estimation of CDF degenerates to the case with only continuous variables, and provide a sufficient condition for the general mixed variables case.