NECESSARY CONDITION FOR THE STRONG UNIFORM CONVERGENCE OF THE NEAREST NEIGHBOUR DENSITY ESTIMATE

Let X_1,X_2,...,X_n be iid. Sample drawn from a population with distribution F, if F have a density, denote it by f. Various methods of estimating f by x_1,X_2,...,X_n, have proposed in the literatures, for example, the kernel estimate and nearest neighhour estimate. The later was introduced by Loftsgarden and Quesenberry1]. The method is as follows: