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

Tsukanov (Theor. Probab. Appl. 26 (1981) 173-177) considers the regression model E(y|Z)=Fp+Zq, D(y|Z)=σ²I_n, where y(n×1) is a vector of measured values,F(n×k) contains the control variables, Z(n×l) contains the observed values, and p(k×1) and q(l×1) are being estimated. Assuming that Z=FL+R, where L(k×l) is non-random, and the rows of R (n×l) are i.i.d. N(0,Σ), we extend Tsukanov's results by (i) computing E(det H_p), where H_p is the covariance matrix of p̂, 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.