Abstract The main Smith selection index objective is to predict the unobservable plant net genetic merit (). When the phenotypic () and genotypic () covariance matrices are estimated, the estimator of this index () is the best predictor of only if the estimator of its vector of coefficients () is unbiased with minimum variance. The expectation and variance of provide an idea of the likely loss of efficiency but those have been an old unsolved problem till now. Assuming that the vector of phenotypic mean values and have joint multivariate normal distribution, we derived the maximum likelihood estimator (MLE) of when is known () and when matrix () is an MLE of . We used the observed Fisher information matrix and the law of total expectation and total variance to show that is a minimum variance unbiased estimator, and we constructed confidence intervals for using the Bonferroni correction for and . Using statistical hypothesis tests, we compared versus and their variances, var () versus var (), assuming that and var () are known. Since the estimator of the index variance () and the prediction error variance () depend on or , and the variance of () depends on or , we compared the estimators of , , and for both cases using statistical hypothesis tests. We did not find significant differences. Therefore, the sampling properties of remain the sampling properties of .
Cerón‐Rojas et al. (Thu,) studied this question.