Uniform in bandwidth consistency of kernel-type function estimators

Abstract

We introduce a general method to prove uniform in bandwidth consistency of kernel-type function estimators. Examples include the kernel density estimator, the Nadaraya-Watson regression estimator and the conditional empirical process. Our results may be useful to establish uniform consistency of data-driven bandwidth kernel-type function estimators.Comment: Published at http://dx.doi.org/10.1214/009053605000000129 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Estimation of density level sets with a given probability content

“…Under Assumption 2, sharp almost-sure convergence rates onf n − f can be established (see, for instance, Giné and Guillou 2002;Einmahl and Mason 2005).…”

“…Under Assumption 2 on the kernel, and since f is bounded under Assumption 1(i), the conditions of Theorem 1 in Einmahl and Mason (2005) are satisfied, implying that ν n ∥f n − Ef n ∥ ∞ → 0 almost surely as n → ∞, where (v n ) is any sequence satisfying v n = o( nh d n / log n). Moreover, under Assumption 1(i), it may be easily shown that ∥Ef n − f ∥ ∞ = O(h 2 n ).…”

Uniform-in-bandwidth nearest-neighbor density estimation

“…Making use of Theorem 1, we readily obtain a description of the limiting behavior of the corresponding NN-estimators. We refer to Einmahl and Mason [16], and Deheuvels and Ouadah [11], for discussions and references on the closely related problem of uniform-in-bandwidth convergence for kernel estimators. To illustrate the sharpness of (5), we set H n = [λ n , λ n ] in Theorem 1, and observe that, whenever {λ n : n ≥ 1} are constants fulfilling, as n → ∞, λ n → 0, and nλ n / log n → ∞,…”

Rates of strong uniform consistency for local least squares kernel regression estimators

“…To obtain the upper bound, the authors use a maximal form of the Bernstein's inequality to estimate the supremum of the deviation on the nodes of a discrete grid and then use the Talagrand's inequality combined with a moment bound to control the difference between the original empirical processes and its values over the nodes of the grid. The main argument of the proof is based on Talagrand's exponential equality [18] for general empirical processes combined with an useful moment inequality for empirical processes indexed by classes of functions of Vapnik-Cervonenkis type (see, e.g., [4], [5], [10] and [11]). Regarding the inner bound part, they apply poissonization techniques in the border of the limiting set σ 2 W (I).…”