Gaussian approximation of local empirical processes indexed by functions

Abstract

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.Mathematics Subject Classiÿcation (1991): 60F15, 62G05

Uniform in bandwidth consistency of kernel regression estimators at a fixed point

“…The work of [2] is based on a notion of a local empirical process indexed by sets. This was further generalized by Einmahl and Mason [7,8] who looked at local empirical processes indexed by functions and who established strong invariance principles for such processes. They inferred LIL type results from the strong invariance principles and this not only for density estimators, but also for the Nadaraya-Watson estimator for the regression function and conditional empirical processes.…”

“…is uniformly bounded in a neighborhood of t, where p > 2 , one needs that {h d n } is at least of order O(n −1 (log n) q ) for some q > 2/(p−2) (see [7,8]). From our main result it will actually follow that in this last case q ≥ 2/(p − 2) is already sufficient.…”

A Test of Non-Identifying Restrictions and Confidence Regions for Partially Identified Parameters

“…Hence Theorem 1.2 of Einmahl and Mason (1997) holds, and sup A∈C b,ln |L n (1 A , ϕ n )| = O a.s. √ ln ln n so that the desired result holds.…”

Gaussian approximation of suprema of empirical processes

“…a "local" empirical process at x (the original definition of the local empirical process in [21] is slightly more general in that h n is replaced by a sequence of bimeasurable functions). With a slight abuse of terminology, we also call the…”

“…For example, [37] considered the Gaussian approximation of W n in the case where Y = R and g(y) = y, but also assumed that the support of Y 1 is bounded. [21] proved in their Theorem 1.1 a weak convergence result for local empirical processes, which, combined with the Skorohod representation and Lemma 4.1 ahead, implies a Gaussian approximation result for W n even when G is not uniformly bounded (but without explicit rates); however, their Theorem 1.1 (and also Theorem 1.2) is tied with the single value of x, that is, x is fixed, since both theorems assume that the "localized" probability measure, localized at a given x, converges (in a suitable sense) to a fixed probability measure (see assumption (F.ii) in [21]). The same comment applies to [22].…”