Optimal designs for ratios of linear combinations in the general linear model
type: Item
instance of: scholarly article
title: Optimal designs for ratios of linear combinations in the general linear model
zbMATH Open document ID: 0595.62069
DOI: 10.1016/0378-3758(86)90144-8
published in: Journal of Statistical Planning and Inference
publication date: 1986-01-01T00:00:00Z
review text: This paper deals with optimal designs for estimating the ratio \(\rho =\theta_ 1/\theta_ 2\) where \(\theta_ 1=\ell '\!_ 1\beta,\) \(\theta_ 2=\ell '\!_ 2\beta,\) \(\beta\) being a (p\(\times 1)\) vector of unknown parameters occurring in the general linear model \[ Y(n\times 1)=X(n\times p)\beta (p\times 1)+\epsilon, \] and \(\ell_ 1\), \(\ell_ 2\) are two independent (p\(\times 1)\) vectors. It is assumed that \(E(\epsilon)=0\) (n\(\times 1)\), \(Var(\epsilon)=\sigma^ 2I_ n,\) and that \(\theta_ i\) \((i=1,2)\) is estimable with \({\hat \theta}{}_ i\) being its BLUE. \({\hat \rho}={\hat \theta}_ 1/{\hat \theta}_ 2\) is the often used maximum likelihood estimator of \(\rho\) when the errors are independent, identically distributed normal random variables. It is shown [see the first author, Commun. Stat., Theory Methods 14, 635-650 (1985; Zbl 0578.62053)] that \(({\hat \rho}_ n-\rho)/\sigma_ n\to N(0,1)\) for normal as well as non-normal errors. The criteria of optimality in this paper are based on minimizing \(\sigma_ n^ 2\) in some sense. The results obtained here are applied to simple linear regression, intersection of two linear regressions, and quadratic regression.
Mathematics Subject Classification ID: 62K05
zbMATH DE Number: 3958481
zbMATH Keywords: ratios of linear combinations
zbMATH Keywords: minimization of asymptotic variance
zbMATH Keywords: optimal designs
zbMATH Keywords: general linear model
zbMATH Keywords: BLUE
zbMATH Keywords: maximum likelihood estimator
zbMATH Keywords: normal
zbMATH Keywords: non-normal errors
zbMATH Keywords: simple linear regression
zbMATH Keywords: intersection of two linear regressions
zbMATH Keywords: quadratic regression
author: John P. Buonaccorsi
author: Hariharan K. Iyer
MaRDI profile type: Publication
full work available at URL: 0378-3758(86)90144-8
OpenAlex ID: W2050939581
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description: scientific article; zbMATH DE number 3958481