A NEW SMOOTHNESS QUANTIFICATION IN KERNEL DENSITY-ESTIMATION

Let X(l),...,X(n) denote a random sample from an unknown univariate distribution with density f. In kernel density estimation, one usually assumes that the density f belongs to some class F, in order to obtain rate of convergence results. The various investigations in the literature differ in the classes of functions F employed, these varying according to smoothness conditions imposed, but typically involve assuming that the k-th derivative of f is bounded uniformly in F (this can be weakened to a Holder condition on the (k-l)-st derivative). In this paper we introduce classes of density functions based on a new smoothness quantification. Our formulation restricts the local behaviour of the density in a neighborhood of each point. Rates of convergence for the mean squared error are established. Relationships with the better known classes, such as the Lipschitz classes, are discussed.