Exact Optimality of Balanced Designs for Minimum Norm Quadratic Unbiased Estimation of Variance Components in one-way Classified data

Abstract

This paper develops an exact theory for the optimality of balanced designs under minimum norm quadratic unbiased estimation of variances components in one-way classified data. A-optimality for a design holds when the trace of its variance covariance matrix for the estimated variance components is minimized. The variance covariance matrices for a random one-way model for two balanced designs different in group sizes are derived and compared. For this type of optimality to hold, the ratio of the variance of groups effects to the error effects is compared with the roots of the quadratic function representing the difference between the traces of the two variance covariance matrices. Whenever the earlier (the ratio) is greater than the latter (the roots) then the A-optimality criteria is satisfied. Comparing the two designs, based on this condition, showed that the design that is larger in groups size meets the A-optimality property.