Cube root asymptotics
type: Item
instance of: scholarly article
title: Cube root asymptotics
zbMATH Open document ID: 0703.62063
author: Elsie Sterbin Gottlieb
published in: The Annals of Statistics
publication date: 1990-01-01T00:00:00Z
review text: This paper is concerned with properties of estimators of a parameter \(\theta\) defined by maximization of processes \[ P_ ng(\cdot,\theta)=(1/n)\sum_{i\leq n}g(X_ i,\theta) \] where \(\{X_ i\}\) is a sequence of independent observations taken from a distribution P and [g(\(\cdot,\theta)\); \(\theta\in \Theta]\) is a class of functions indexed by a subset \(\Theta\) of \(R^ d\). Let \([\theta_ n]\) be a sequence of estimators for which \[ P_ ng(\cdot,\theta_ n)\geq \sup_{\theta \in \Theta}P_ ng(\cdot,\theta)-o_ p(n^{-2/3}). \] Suppose \(\theta_ n\) converges in probability to the unique \(\theta_ 0\) that maximizes Pg(\(\cdot,\theta)\) and \(\theta_ 0\) is an interior point of \(\Theta\). Assume that Pg(\(\cdot,\theta)\) is twice differentiable with second derivative matrix -V at \(\theta_ 0\), and \[ H(s,t)=\lim_{A\to \infty}A Pg(\cdot,\theta_ 0+s/A)g(\cdot,\theta_ 0+t/A) \] exists for each s,t in \(R^ d\). Then under some additional regularity conditions, the process \(n^{2/3}P_ ng(\cdot,\theta_ 0+tn^{-1/3})\) converges in distribution to a Gaussian process Z(t) with continuous sample paths, expected value \(-t'Vt/2\), and covariance kernel H. If V is positive definite and Z has nondegenerate increments, then \(n^{1/3}(\theta_ n-\theta_ 0)\) converges in distribution to the (almost surely unique) random vector that maximizes Z. Various examples are given.
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 62G99
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62M99
zbMATH DE Number: 4153723
zbMATH Keywords: empirical processes
zbMATH Keywords: empirical measure
zbMATH Keywords: limiting Gaussian process
zbMATH Keywords: least median of squares estimator
zbMATH Keywords: maximum likelihood estimator
zbMATH Keywords: functional central limit theorem
zbMATH Keywords: almost-sure representation
zbMATH Keywords: empirical process
zbMATH Keywords: VC class
zbMATH Keywords: Brownian motion with quadratic drift
zbMATH Keywords: maximum of a Gaussian process
zbMATH Keywords: shorth
zbMATH Keywords: maximum score estimator
zbMATH Keywords: monotone density
zbMATH Keywords: continuous sample paths
zbMATH Keywords: examples
reviewed by: Lionel Weiss
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176347498
OpenAlex ID: W1995642533
DOI: 10.1214/AOS/1176347498
Recommended article: Cubics with a Rational Root
Recommended article: Q4472971
Recommended article: Formulas for cube roots in \(\mathbb F_{3^m}\)
Recommended article: Q5658135
Recommended article: Q4243007
Recommended article: Approximation of the smallest root of a cubic equation
Recommended article: Asymptotics and formulas for cubic exponential sums
Recommended article: Q3575912
description: scientific article; zbMATH DE number 4153723