The Annals of Probability
- Ann. Probab.
- Volume 31, Number 2 (2003), 719-768.
The $\bm{L}_\mathbf{1}$-norm density estimator process
Evarist Giné, David M. Mason, and Andrei Yu. Zaitsev
Full-text: Open access
Abstract
The notion of an $L_{1}$-norm density estimator process indexed by a class of kernels is introduced. Then a functional central limit theorem and a Glivenko--Cantelli theorem are established for this process. While assembling the necessary machinery to prove these results, a body of Poissonization techniques and restricted chaining methods is developed, which is useful for studying weak convergence of general processes indexed by a class of functions. None of the theorems imposes any condition at all on the underlying Lebesgue density $f$. Also, somewhat unexpectedly, the distribution of the limiting Gaussian process does not depend on $f$.
Article information
Source
Ann. Probab. Volume 31, Number 2 (2003), 719-768.
Dates
First available in Project Euclid: 24 March 2003
Permanent link to this document
https://projecteuclid.org/euclid.aop/1048516534
Digital Object Identifier
doi:10.1214/aop/1048516534
Mathematical Reviews number (MathSciNet)
MR1964947
Zentralblatt MATH identifier
1031.62026
Subjects
Primary: 60F05: Central limit and other weak theorems 60F15: Strong theorems 60F17: Functional limit theorems; invariance principles 62G07: Density estimation
Keywords
Kernel density function estimator $L_{1}$-norm central limit theorem weak convergeance Poissonization entropy
Citation
Giné, Evarist; Mason, David M.; Zaitsev, Andrei Yu. The $\bm{L}_\mathbf{1}$-norm density estimator process. Ann. Probab. 31 (2003), no. 2, 719--768. doi:10.1214/aop/1048516534. https://projecteuclid.org/euclid.aop/1048516534
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- STORRS, CONNECTICUT 06269-3009 E-MAIL: gine@uconnvm.uconn.edu D. M. MASON DEPARTMENT OF FOOD AND RESOURCE ECONOMICS 206 TOWNSEND HALL UNIVERSITY OF DELAWARE
- NEWARK, DELAWARE 19717 E-MAIL: davidm@udel.edu A. YU. ZAITSEV LABORATORY OF STATISTICAL METHODS ST. PETERSBURG BRANCH OF THE STEKLOV MATHEMATICAL INSTITUTE 27 FONTANKA ST. PETERSBURG 191011 RUSSIA E-MAIL: zaitsev@pdmi.ras.ru
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