Kernel Estimation of Multivariate Conditional Distributions

We consider the problem of estimating conditional probability distributions that are multivariate in both the conditioned and conditioning variable sets. This is an extension of Hall, Racine, and Li (forthcoming), who considered the case of a univariate conditioned variable but who also considered the more general case of both irrelevant and relevant conditioning variables. Following Hall et al. (forthcoming), we use the kernel method with the smoothing parameters selected from the cross-validated minimization of a weighted integrated squared error of the kernel estimator. We derive the rate of convergence of the smoothing parameters to some non-stochastic optimal smoothing parameter values, and establish the asymptotic normal distribution of the resulting nonparametric conditional probability (density) estimator. Simulations show that the proposed method performs quite well with a mixture of categorical and continuous variables.