The Law of the Iterated Logarithm for the Multivariate Nearest Neighbor Density Estimators

We consider estimation of a multivariate probability density function f(x) by kernel type nearest neighbor (nn) estimators gn(x). The development of nn density estimation theory has had a rich history since Loftsgaarden and Quesenberry proposed the idea in 1965. In particular, there is a vast literature on convergence properties of gn(x) to f(x). For statistical purposes, however, it is of importance to study also the speed of almost sure convergence. For pointwise estimation, this problem appears to have received no attention in the literature. The aim of the present paper is to obtain sharp pointwise rates of strong consistency by establishing a law of the iterated logarithm for this class of estimators. We also study the local estimation of a density function based on censored data by the kernel smoothing method using a nearest neighbor approach and derive a law of the iterated logarithm.