Symmetrized Multivariate k-NN Estimators

In this article, we propose a symmetrized multivariate k-NN estimator for the conditional mean and for the conditional distribution function. We establish consistency and asymptotic normality of each estimator. For the estimator of the conditional distribution function, we also establish the weak convergence of the conditional empirical process to a Gaussian process. Compared with the corresponding kernel estimators, the asymptotic distributions of our k-NN estimators do not depend on the existence of the marginal probability density functions of the covariate vector. A small simulation study compares the finite sample performance of our symmetrized multivariate k-NN estimator with the Nadaraya-Watson kernel estimator for the conditional mean.