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 p [greater, double equals] 0. It is shown that f(p) are uniformly strongly consistent estimators of f(p), the pth order derivative of f, if and...

On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators fn(p) of f(p), where f is a density of X w.r.t. a [sigma]-finite measure dominated by the Lebesgue measure on Rm, P = (p1,...,pm), pj = 0, fixed integers, and for x =...

Let Xj = (X1j ,..., Xpj), j = 1,..., n be n independent random vectors. For x = (x1 ,..., xp) in Rp and for [alpha] in [0, 1], let Fj(x) = [alpha]I(X1j < x1 ,..., Xpj < xp) + (1 - [alpha]) I(X1j <= x1 ,..., Xpj <= xp), where I(A) is the indicator random variable of the event A. Let Fj(x) = E(Fj(x)) and Dn = supx, [alpha] max1 <= N <= n [Sigma]0n(Fj(x) - Fj(x)). It is shown that P[Dn >= L] < 4pL exp{-2(L2n-1 - 1)} for each positive integer n and for all L2 >= n; and, as n --> [infinity], Dn = 0((nlogn)1/2) with probability one.