Testing optimality of experimental designs for a regression model with random variables

Tsukanov (Theor. Probab. Appl. 26 (1981) 173–177) considers the regression model $\text{E(}\text{y}\text{|}\text{Z}\text{)=}\text{Fp}\text{+}\text{Zq}$, $\text{D(}\text{y}\text{|}\text{Z}\text{)=σ}{}^2\text{I}{}_{\mathrm{n}}$, where $\text{y}\text{(n×1)}$ is a vector of measured values,$\text{F}\text{(n×k)}$ contains the control variables, $\text{Z}\text{(n×l)}$ contains the observed values, and $\text{p}\text{(k×1)}$ and $\text{q}\text{(l×1)}$ are being estimated. Assuming that $\text{Z}\text{=}\text{FL}\text{+}\text{R}$, where $\text{L}\text{(k×l)}$ is non-random, and the rows of $\text{R}\text{(n×l)}$ are i.i.d. $\text{N(}\text{0}\text{,Σ)}$, we extend Tsukanov's results by (i) computing E(detH_p), where H_p is the covariance matrix of p?, the l.s.e. of p, (ii) considering ‘optimality in the mean’ for the largest root criterion, (iii) discussing these equations when the matrix R has a left-spherical distribution.