Nonparametric estimation of mixed partial derivatives of a multivariate density

Abstract

On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators f n ( p) of f ( p) , where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on R m , p = ( p 1,…, p m ), p j ≥ 0, fixed integers, and for x = ( x 1,…, x m ) in R m , f (p)(x) = ∂ p 1+…+p m f (x)/(∂ p 1 x 1 … ∂ p m x m ). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of f n ( p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of f n ( p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.