Testing optimality of experimental designs for a regression model with random variables (Q1084811)

\textit{A. V. Tsukanov} [Teor. Veroyatn. Primen. 26, 178-182 (1981; Zbl 0455.62062); English translation in Theory Probab. Appl. 26, 173-177 (1981)] considers the regression model \(E(y| Z)=Fp+Zq\), \(D(y| Z)=\sigma^ 2I_ n\), where y(n\(\times 1)\) is a vector of measured values, F(n\(\times k)\) contains the control variables, Z(n\(\times l)\) contains the observed values, and p(k\(\times 1)\) and q(l\(\times 1)\) are being estimated. Assuming that \(Z=FL+R\), where L(k\(\times l)\) is non-random, and the rows of R(n\(\times l)\) are i.i.d. N(0,\(\Sigma)\), we extend Tsukanov's results by (i) computing E(det \(H_ p)\), where \(H_ p\) is the covariance matrix of \(\hat p,\) 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.