Asymptotic Properties of Certain Measures of Deviation for Kernel-Type Nonparametric Estimators of Probability Densities

Abstract

1. Let X₁, X₂, ..., X_n be independent identically distributed one dimensional random variables possessing unknown probability density function f(x). We assume that the required density f(x) belongs to the space L₂ (− ∞, ∞) of functions which are square integrable with respect to the Lebesgue measure and consider the methods of empirical approximations to this density — when the errors are measured in the L(− ∞, ∞) metric — of the form

$$\mathop f\nolimits_n (x, \mathop a\nolimits_n ) = \mathop n\nolimits^{ - 1} \mathop a\nolimits_n \sum\limits_{i = i}^n {} K(\mathop a\nolimits_n (x - \mathop x\nolimits_i )) ,$$

(1.1)

Here K(x) is a function belonging to L₂ (− ∞, ∞) which satisfies the regularity conditions H_s stipulated below and a_n is a sequence of positive numbers converging to infinity but fulfilling the condition a_n = o(n) as n ➛ ∞.