The Annals of Statistics

Cube Root Asymptotics

Jeankyung Kim and David Pollard

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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.

Article information

Ann. Statist., Volume 18, Number 1 (1990), 191-219.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G15: Gaussian processes 62G99: None of the above, but in this section

Functional central limit theorem almost-sure representation empirical process VC class Brownian motion with quadratic drift maximum of a Gaussian process shorth least median of squares maximum score estimator monotone density


Kim, Jeankyung; Pollard, David. Cube Root Asymptotics. Ann. Statist. 18 (1990), no. 1, 191--219. doi:10.1214/aos/1176347498.

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