Estimation of Variance Components Under a Selection Model

A procedure for the estimation of variance components free of bias due to selection on a random vector correlated with the observation vector is presented.The estimators are based on restricted maximum likelihood methods for unselected situations and expectations derived under the pretense that a priori values are equal to true values.In the case of no selection, the formulas reduce to the usual formulas for restricted maximum likelihood.The properties of estimators under a selection model were not derived, because after selection, multivariate normality may no longer be assumed.Applications to particular selection problems in dairy cattle are discussed.