Estimation of prediction error variances via Monte Carlo sampling methods using different formulations of the prediction error variance

In quantitative genetics the prediction error variance-covariance matrix is central to the calculation of accuracies of estimated breeding values ( u ^ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCyDauNbaKaaaaa@2D5C@ ) [e.g. [1]], to REML algorithms for the estimation of variance components [2], to methods which restrict the variance of response to selection [3], and can be used to explore trends in Mendelian sampling deviations over time [4]. The mixed model equations (MME) for most national genetic evaluations range from 100,000 to 20,000,000 equations and inversion of systems of equations of this size is generally not possible because of their magnitude or because of loss of numerical precision [5]. Methods that approximate the prediction error variances (PEV) and calculate the accuracy of u ^ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafeyDauNbaKaaaaa@2D56@ provide biased estimates in some circumstances by ignoring certain information [e.g. [6]]. Variance components upon which genetic evaluations of large populations are based are generally estimated using reduced data sets. The use of reduced data sets may create bias in the estimates as REML only provides unbiased estimates of variance components when all the data on which selection has taken place is included in the analysis [7]. Variance of response to selection is generally not controlled in breeding programs although it might be a risk to them [3].Approximations of the PEV without needing to invert the coefficient matrix or to delete data, can be obtained by comparing Monte Carlo samples of the data and suc