Equivalence of uniform asymptotic unbiasedness, mean square and strong consistencies of recursive estimates of a density and its p-th derivative

Abstract

Menon, Prasad and Singh have given nonparametric recursive kernel estimators of a probability density function f and its p-th order derivative f ( p) based on a random sample of size n from f. Denoting the estimates of f ( p) (where f (0) is simply the p.d.f. f) by f n, p they have shown that if f ( p) is bounded and uniformly continuous, then their recursive estimators f n, p are uniformly mean square as well as uniformly strongly consistent for f ( p) . In this note we will prove the converse. That is, we show that for uniform strong consistency or for uniform mean square consistency of f n, p it is necessary that f ( p) be bounded and uniformly continuous, whether it is the case of p = 0 or p ≥ 1. Under certain conditions on the window-width function, it is shown that each of the properties of uniform asymptotic unbiasedness, uniform mean square consistency and uniform strong consistency of f n, p is equivalent to uniform continuity of f ( p) along with its boundedness.