A LAW OF THE ITERATED LOGARITHM FOR NEAREST NEIGHBOR ESTIMATION OF MULTIVARIATE DENSITY FUNCTION

Abstract

Let X be a d-dimensional random vector with unknown density function f(z) =f(z1, …, zd), and let fn be teh nearest neighbor estimator of f proposed by Loftsgaarden and Quesenberry (1965). In this paper, we established the law of the iterated logarithm of fn for general case of d≥1, which gives the exact pointwise strong convergence rate of fn.