Cube Root Asymptotics

Kim, Jeankyung ; Pollard, David

Ann. Statist., Tome 18 (1990) no. 1, p. 191-219 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1990-03-14
Classification:  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,  60F17,  60G15,  62G99