Maximum likelihood sampling properties of the estimator of the Smith selection index vector of coefficients

Abstract

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