On necessary and sufficient conditions for uniform strong consistency of estimators of a density and its derivatives

Based on a random sample from a population with (unknown) probability density f, this note exhibits a class of statistics f^(p) for each fixed integer $\text{p ≧ 0}$. It is shown that f^(p) are uniformly strongly consistent estimators of f^(p), the pth order derivative of f, if and only iff^(p)is bounded and uniformly continuous.