Symmetrized Multivariate k-NN Estimators

Abstract

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.

Keywords:

• Conditional empirical process
• Conditional quantile process
• Copula
• Local oscillation of empirical processes

JEL Classification:

• C13
• C14
• C21

ACKNOWLEDGMENT

We thank Qi Li and two anonymous referees for helpful suggestions that have improved the paper. Ruixuan Liu expresses his deep indebtedness to Prof. Ullah when he left U.C. Riverside.

Notes

The global bandwidth is chosen mainly to ease the computational burden, since it is one built-in choice in Loader's locfit package. More computationaly intensive cross-validated bandwidth could be used as well.

The result based on Triweight kernel is almost identical and hence omitted.