Optimal designs for ratios of linear combinations in the general linear model (Q1077850)

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.