Mise of kernel estimates of a density and its derivatives

Abstract

For an integer p ⩾ 0, Singh has considered a class of kernel estimators ƒ∼ (p) of the pth order derivative ƒ (p) of a density ƒ and showed how specializations of some of the results there improve the corresponding existing results. In this paper these improved estimators are examined on a global measure of quality of an estimator, namely, the mean integrated square error (MISE) behavior. An upper bound, which can not be tightened any further for a wide class of kernels, is obtained for MISE ( ƒ∼ (p) ). The exact asymptotic value for the same is also obtained. Under two alternative conditions, weaker than those assumed for the two results mentioned above, convergence of MISE ( ƒ∼ (p) ) to zero is proved. Specializations of some of the results here improve the corresponding existing results by weakening the conditions, sharpening the rates of convergence or both.