Multivariate data-driven k-NN function estimation

In Bhattacharya and Mack (Ann. Statist. 15 (1987), 976-994), it was shown (among other things) that adapting for the optimal choice of k in univariate k-nearest neighbor density and regression estimation is feasible using weak convergence techniques. We now show that the same holds true for the multivariate case. Our results parallel Krieger and Pickands (Ann. Statist. 9 (1981), 1066-1078) and Mack and Müller (J. Multivariate Anal. 23 (1987), 169-182) for adaptive multivariate kernel density, respectively, regression, estimation.