Estimation problems for samples with measurement errors
type: Item
instance of: scholarly article
title: Estimation problems for samples with measurement errors
zbMATH Open document ID: 0591.62025
author: Wolfgang Stadje
published in: The Annals of Statistics
publication date: 1985-01-01T00:00:00Z
review text: For \(x\in R\), let \(N_{\alpha}(x)=m\alpha\) iff \(x\in (\alpha m-\alpha /2\), \(\alpha m+\alpha /2]\). Also let \(X_ 1,...,X_ n\) be a random sample from the pdf \(f_ X(x)\), and set \[ \bar N_{\alpha}=(1/n)\sum^{n}_{i=1}N_{\alpha}(X_ i)\quad and\quad S^ 2_{\alpha}=[1/(n-1)]\sum^{n}_{i=1}[N_{\alpha}(X_ i)-\bar N_{\alpha}]^ 2. \] The author mainly studies the asymptotic properties of \(\bar N_{\alpha_ n}\) and \(S^ 2_{\alpha_ n}\), as \(n\to \infty.\) For example, if \(E(X^ 2)<\infty\), \(E(e^{itX})=o(| t|^{-k})\) (as \(| t| \to \infty)\) for some \(k\in {\mathbb{N}}\) and \(\alpha_ n=O(n^{-1/(2k+2)})\), or X is distributed as \(N(\theta,\sigma^ 2)\) and \(\alpha_ n\leq 2\pi \sigma (\log n)^{- 1/2}\), the author shows that \(n^{1/2}[\bar N_{\alpha_ n}-E(X)]\) is asymptotically normal. The author also considers problems of truncation and general maximum likelihood estimation from discrete scale measurements.
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 3949490
zbMATH Keywords: asymptotic normality
zbMATH Keywords: asymptotic unbiasedness
zbMATH Keywords: efficiency
zbMATH Keywords: measurement errors
zbMATH Keywords: truncation
zbMATH Keywords: maximum likelihood estimation
zbMATH Keywords: discrete scale measurements
reviewed by: Andreas N. Philippou
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176349757
OpenAlex ID: W2021845417
DOI: 10.1214/AOS/1176349757
description: scientific article; zbMATH DE number 3949490