Some statistical properties of a genetic selection index

The genetic selection index proposed by Smith ( 1936) is an optimal linear index involving parametric weighting coefficients. In practical applications, these weights have to be estimated using sample data. The statistical properties of the resulting index, which will be called the estimated index, are still in doubt although some approximations to the moments of estimated weights and the expected gain due to selection have been obtained by Bartlett (1939) and Nanda (1949). The framework of the estimation problem for an additive normal vector of observations can be summarized as follows. Let p be a vector of q observable variables (phenotypes), g a vector of q non-observable worth variables (genotypes), and e a vector of q observation errors (environment) where p= ,u+g+e,