Nonparametric estimation of mixed partial derivatives of a multivariate density

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 = (x1,...,xm) in Rm, f(p)(x) = [not partial differential]p1+...+pm f(x)/([not partial differential]p1x1 ... [not partial differential]pmxm). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of fn(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of fn(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.