Mixed-rates asymptotics

The Annals of Statistics · 2008 · Volume 36 · Issue 1

Abstract

A general method is presented for deriving the limiting behavior of estimators that are defined as the values of parameters optimizing an empirical criterion function. The asymptotic behavior of such estimators is typically deduced from uniform limit theorems for rescaled and reparametrized criterion functions. The new method can handle cases where the standard approach does not yield the complete limiting behavior of the estimator. The asymptotic analysis depends on a decomposition of criterion functions into sums of components with different rescalings. The method is explained by examples from Lasso-type estimation, k-means clustering, Shorth estimation and partial linear models.

Keywords

60F17, 62F12, Empirical processes, K-means, Lasso, Limiting distribution, M-estimators, Nonstandard asymptotics, Partial splines, rates of convergence, shorth, singular quadratic approximation

Enable JavaScript for citation tools and additional paper interactions.