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2008 A limited in bandwidth uniformity for the functional limit law of the increments of the empirical process
Davit Varron
Electron. J. Statist. 2: 1043-1064 (2008). DOI: 10.1214/08-EJS193

Abstract

Consider the following local empirical process indexed by $K\in \mathcal{G}$, for fixed h>0 and zd: $$G_{n}(K,h,z):=\sum_{i=1}^{n}K\Bigl(\frac{Z_{i}-z}{h^{1/d}}\Big)-\mathbb{E}\Bigl(K\Bigl(\frac{Z_{i}-z}{h^{1/d}}\Big)\Big),$$ where the Zi are i.i.d. on ℝd. We provide an extension of a result of Mason (2004). Namely, under mild conditions on $\mathcal{G}$ and on the law of Z1, we establish a uniform functional limit law for the collections of processes $\bigl\{G_{n}(\cdot,h_{n},z),\;z\in H,\;h\in [h_{n},\mathfrak{h}_{n}]\big\}$, where Hd is a compact set with nonempty interior and where hn and $\mathfrak{h}_{n}$ satisfy the Csörgő-Révész-Stute conditions.

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Davit Varron. "A limited in bandwidth uniformity for the functional limit law of the increments of the empirical process." Electron. J. Statist. 2 1043 - 1064, 2008. https://doi.org/10.1214/08-EJS193

Information

Published: 2008
First available in Project Euclid: 13 November 2008

zbMATH: 1320.60091
MathSciNet: MR2460857
Digital Object Identifier: 10.1214/08-EJS193

Subjects:
Primary: 60F15 , 60F17
Secondary: 62G07

Keywords: Density estimation , Empricial processes , Functional limit theorems , LaTeXe , strong theorems

Rights: Copyright © 2008 The Institute of Mathematical Statistics and the Bernoulli Society

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