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March, 1990 Cube Root Asymptotics
Jeankyung Kim, David Pollard
Ann. Statist. 18(1): 191-219 (March, 1990). DOI: 10.1214/aos/1176347498


We establish a new functional central limit theorem for empirical processes indexed by classes of functions. In a neighborhood of a fixed parameter point, an $n^{-1/3}$ rescaling of the parameter is compensated for by an $n^{2/3}$ rescaling of the empirical measure, resulting in a limiting Gaussian process. By means of a modified continuous mapping theorem for the location of the maximizing value, we deduce limit theorems for several statistics defined by maximization or constrained minimization of a process derived from the empirical measure. These statistics include the short, Rousseeuw's least median of squares estimator, Manski's maximum score estimator, and the maximum likelihood estimator for a monotone density. The limit theory depends on a simple new sufficient condition for a Gaussian process to achieve its maximum almost surely at a unique point.


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Jeankyung Kim. David Pollard. "Cube Root Asymptotics." Ann. Statist. 18 (1) 191 - 219, March, 1990.


Published: March, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0703.62063
MathSciNet: MR1041391
Digital Object Identifier: 10.1214/aos/1176347498

Primary: 60F17
Secondary: 60G15, 62G99

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 1 • March, 1990
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