Nonnegative minimum biased quadratic estimation in the linear regression models
type: Item
instance of: scholarly article
title: Nonnegative minimum biased quadratic estimation in the linear regression models
zbMATH Open document ID: 0844.62052
author: Stanisław Gnot
author: Götz Trenkler
author: Roman Zmyślony
published in: Journal of Multivariate Analysis
publication date: 1996-09-01T00:00:00Z
review text: Let us consider a linear model \(M\{y, X\beta, \sigma^2 I\}\), where \(y\) is an \(n \times 1\) normally distributed vector of observations, with the expectation vector \(E(y) = X \beta\) and the covariance matrix \(\text{Cov} (y) = \sigma^2 I\). The nonstochastic \(n \times p\) matrix \(X\) is known and of rank \(p - q\), \(0 \leq q < p < n\), \(\beta\) is a \(p \times 1\) vector of unknown parameters, while \(\sigma^2 > 0\) is the unknown variance of the disturbances. We consider the problem of nonnegative estimation of \(\beta' H\beta + h\sigma^2\). Here \(H\) is a nonnegative definite matrix, while \(h\) is a nonnegative scalar. The problem arises, for instance, if we want to estimate accuracy of linear estimators \(Ly\) of \(\beta\) by means of the total mean squared error (TMSE), which has the structure \(\beta' H\beta + h \sigma^2\) and it is given by \[ \text{TMSE} (Ly) = E[(Ly - \beta)' (Ly - \beta)] = \beta' (X' L' - I) (X' L' - I)' \beta + \sigma^2 \text{tr} \{LL'\}. \] An iterative procedure for the nonnegative minimum biased quadratic estimator is described. Moreover, in the case that \(H\) and \(X' X\) commute, an explicit formula for this estimator is given. Admissibility of the estimator is proved. The results are applied to nonnegative estimation of the total mean squared error of a linear biased estimator.
Mathematics Subject Classification ID: 62J05
Mathematics Subject Classification ID: 62H12
zbMATH DE Number: 797144
zbMATH Keywords: one-way classification model
zbMATH Keywords: admissibility
zbMATH Keywords: linear model
zbMATH Keywords: iterative procedure
zbMATH Keywords: nonnegative minimum biased quadratic estimator
zbMATH Keywords: nonnegative estimation
zbMATH Keywords: total mean squared error
zbMATH Keywords: linear biased estimate
MaRDI profile type: Publication
full work available at URL: jmva.1995.1047
OpenAlex ID: W2048031614
DOI: 10.1006/JMVA.1995.1047
description: scientific article; zbMATH DE number 797144