Strong convergence of distributions of estimators
type: Item
instance of: scholarly article
title: Strong convergence of distributions of estimators
zbMATH Open document ID: 0637.62027
author: P. Jeganathan
published in: The Annals of Statistics
publication date: 1987-01-01T00:00:00Z
review text: Let the parameter space \(\theta\) be an open subset of R k and \(\{T_ n\}^ a \)sequence of estimators of \(\vartheta\in \theta\). Assuming that the locally asymptotically normal condition is satisfied at \(\vartheta_ 0\in \theta\), the author obtained a sufficient condition which assures that for all \(\alpha >0\lim_{n\to \infty}\sup_{| h| \leq \alpha}\sup_{| f| \leq 1}| E[f(T\quad *\quad_ n-h| Q_{h,n})]-\int f(x)L(dx)| =0\) where \(T\) \(*_ n=\delta_ n^{- 1}(T_ n-\vartheta_ 0)\), \(\delta_ n\) is some positive definite matrix, \(h\in R\) k, f is a Borel measurable function and L denotes the limit distribution of \(T\) \(*_ n\) under \(Q_{0,n}.\) This convergence is weaker than the convergence of densities but sufficient for many statistical purposes. The method of proof used in this paper improves \textit{D. D. Boos}' method, Ann. Stat. 13, 423-427 (1985; Zbl 0567.62012) and seems to be applicable to establish such convergence of densities.
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 4039013
zbMATH Keywords: convergence in law of estimators
zbMATH Keywords: strong convergence
zbMATH Keywords: local asymptotic normality
zbMATH Keywords: asymptotic equivariance
zbMATH Keywords: LAN condition
reviewed by: Ken-ichi Yoshihara
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176350619
OpenAlex ID: W2058973834
DOI: 10.1214/AOS/1176350619
description: scientific article; zbMATH DE number 4039013