Local Empirical Processes Research Articles

Gaussian approximation of local empirical processes indexed by functions

An extended notion of a local empirical process indexed by functions is introduced, which includes kernel density and regression function estimators and the conditional empirical process as special cases. Under suitable regularity conditions a central limit theorem and a strong approximation by a sequence of Gaussian processes are established for such processes. A compact law of the iterated logarithm (LIL) is then inferred from the corresponding LIL for the approximating sequence of Gaussian processes. A number of statistical applications of our results are indicated.

Nonstandard local empirical processes indexed by sets

Abstract Let μn denote the empirical measure based on the first n observations from a sequence of independent random variables with common distribution μ in . Let be a class of Borel subsets of . For any and sequence k(n) with (log log n)−1k(n) → c ϵ (0, ∞), let , , be the empirical process at indexed by . Under suitable assumptions on μ and , we obtain functional laws of the iterated logarithm for .

Functional Laws of the Iterated Logarithm for Local Empirical Processes Indexed by Sets

We introduce the notion of a multivariate local (and tail) empirical process indexed by sets and establish a number of functionals laws of the iterated logarithm for such processes. This leads to a unified methodology to study the almost sure behavior of various statistics which are local functionals of the empirical distribution. Such statistics include density estimators and the Bahadur-Kiefer representation.