Estimation for the nonlinear functional relationship
type: Item
instance of: scholarly article
title: Estimation for the nonlinear functional relationship
zbMATH Open document ID: 0669.62046
author: Yasuo Amemiya
author: Wayne A. Fuller
published in: The Annals of Statistics
publication date: 1988-01-01T00:00:00Z
review text: Let \(\{b_ n\}^{\infty}_{n=1}\) and \(\{a_ n\}^{\infty}_{n=1}\) be sequences of positive real numbers such that \(a_ n\cdot b_ n=n\). Let \[ f(z^ 0_ t,\beta^ 0)=0,\quad t=1,2,...,b_ n \] be a functional relationship, where \(z^ 0_ t\) are unobservable fixed vectors belonging to a parameter space \(\Gamma \subset R^ p\), \(\beta^ 0\in \Omega \subset R^ k\) is a \(1\times k\) vector of unknown parameters, \[ f(z,\beta):\quad \Gamma \times \Omega \to R^ 1. \] The observations are the p-dimensional vectors \[ Z_{nt}=z^ 0_ t+\epsilon_{nt},\quad t=1,...,b_ n, \] where \(\epsilon_{nt}\) are i.i.d. r.v. with mean zero and covariance matrix \(\Sigma_ n=a_ n^{- 1}\Phi\), where \(\Phi >0\) is a fixed matrix. The maximum likelihood estimators \({\hat \beta}\) and \(\hat z_ t\) are the values of \(\beta\) in \(\Omega\) and \(z_ t\) in \(\Gamma\) that minimize \[ \sum^{b_ n}_{t=1}(Z_{nt}-z_ t)\Sigma_ n^{-1}(Z_{nt}-z_ t)' \] subject to \(f(z_ t,\beta)=0\), \(t=1,...,b_ n\). Conditions for consistency and asymptotic normality of \({\hat \beta}\) and \(\hat z_ t\) are investigated. Modifications of the maximum likelihood estimators (bias-adjusted estimators) are also considered.
Mathematics Subject Classification ID: 62J02
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62H12
zbMATH DE Number: 4096602
zbMATH Keywords: nonlinear implicit relationship
zbMATH Keywords: measurement errors
zbMATH Keywords: asymptotic bias
zbMATH Keywords: maximum likelihood estimators
zbMATH Keywords: consistency
zbMATH Keywords: asymptotic normality
zbMATH Keywords: bias- adjusted estimators
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176350696
OpenAlex ID: W2007727338
DOI: 10.1214/AOS/1176350696
description: scientific article; zbMATH DE number 4096602