Statistical Sampling Properties of the Coefficients of Three Phenotypic Selection Indices
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The aim of the Smith phenotypic selection index (SPSI), the restricted phenotypic selection index (RPSI), and the predetermined proportional gains phenotypic selection index (PPG-PSI) is to maximize the response to selection and provide the breeder with an objective rule for evaluating and selecting several traits. When the phenotypic and genotypic variances and covariances are known, these three indices are the best linear predictors. When these parameters are estimated, the three indices will be optimal only if the estimators of the index weights are unbiased and have minimal variance. There are many methods for determining the sampling properties of the SPSI but there is no method for determining the sampling properties of RPSI and PPG-PSI coefficients. Using the canonical correlation theory, we proposed an asymptotic method for determining the statistical sampling properties of the estimators of the coefficients of the three phenotypic selection indices. We showed that under some assumptions, the sampling properties of the RPSI and PPG-PSI coefficient estimators could be obtained using the sampling properties of the SPSI coefficient estimator. We validated the theoretical results using two real datasets. The theoretical and numerical results indicated that the three estimators of the weights for the three indices were unbiased with minimal variances. We concluded that when the number of genotypes is large, the proposed method could be used to find the sampling properties of the coefficient of the three indices.
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