Nonparametric Estimation of Derivatives of Average of μ-Densities with Convergence rates and Applications

Let Xⱼ, 1 j n, be independent random variables with densities fⱼ with respect to a σ-finite measure μ on the real line R. Let fⱼ( ν ) be the νth derivative of fⱼ. With a given integer p 0, estimations of f( p ) = n- 1 ∑₁ⁿ fⱼ( p ) by use of X₁ , ⋯ ,Xₙ are considered, and their applications to quite a few statistical problems are pointed out. For a subset S of R sufficient conditions for asymptotic unbiasedness (asy. u.), almost sure (a.s.) and mean square (ms.) consistencies (of our estimators) each uniform on S, and each with and without rates, are given. When only finitely many fⱼ are distinct, necessary conditions for asy. u., ms. consistency or a.s. consistency, uniform on ( a,∞ ) for an a - ∞, are given. When fⱼ ≡ f are Lebesgue densities, it turns out that the estimators of f( p ) are asy. unbiased, ms. consistent or a.s. consistent, uniformly on ( a,∞ ) for an a - ∞if and only if f( p ) is uniformly right continuous there. It is observed that the rates obtained here for asy. u. and ms. consistencies cannot be improved further by any kernel estimators without making stronger assumptions.