Equivalence of uniform asymptotic unbiasedness, mean square and strong consistencies of recursive estimates of a density and its p-th derivative (Q1084786)

\textit{V. V. Menon}, \textit{R. S. Singh} and the author [ibid. 9, 73-82 (1984; Zbl 0535.62034)] 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 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 \(p\geq 1\). 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 for \(f^{(p)}\) along with its boundedness.