## The Annals of Statistics

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

### Cube Root Asymptotics

Jeankyung Kim and David Pollard

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1176347498

**Digital Object Identifier**

doi:10.1214/aos/1176347498

**Mathematical Reviews number (MathSciNet)**

MR1041391

**Zentralblatt MATH identifier**

0703.62063

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles

Secondary: 60G15: Gaussian processes 62G99: None of the above, but in this section

**Keywords**

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

#### Citation

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