Slanger W D
Department of Animal and Range Sciences, North Dakota State University, Fargo 58105, USA.
J Anim Sci. 1996 Nov;74(11):2577-85. doi: 10.2527/1996.74112577x.
Variances of quadratic estimators of (co)variances are functions of the numeric values of the (co)variance parameters being estimated. This situation makes estimation of (co)variances problematical. Uniformly best quadratic, unbiased estimators exist for balanced designs but not for unbalanced designs. This article tackles the problem by providing explicit quadratic estimators of (co)variances that are uniformly best in the sense that they are uniformly minimum variance, unbiased to the maximum extent possible over the entire range of possible parameter values of the (co)variances being estimated. This was accomplished by determining the restrictions on the elements of the matrix of the quadratic-form matrix necessary to satisfy the Lehmann-Scheffé criterion for uniformly minimum variance, unbiased estimation and then solving the resulting linear equations via the principle of least squares. The context is any mixed linear model, and the approach does not require that there be equal numbers of observations in the case of multivariate data. A detailed development of the method is given. That the procedure is completely general is discussed. A modification that forces unbiasedness is presented. An example with a three-variance-component model is provided and results discussed. A miscellaneous section discusses, among other topics, how this method can be used to compare other (co)variance component estimation procedures. The final section illustrates how the method handles multivariate situations (i.e., models with both variances and covariances) by detailing the expressions involved with the bivariate model.
（协）方差二次估计量的方差是被估计的（协）方差参数数值的函数。这种情况使得（协）方差的估计存在问题。对于平衡设计存在一致最优的二次无偏估计量，但对于非平衡设计则不存在。本文通过提供明确的（协）方差二次估计量来解决该问题，这些估计量在以下意义上是一致最优的：它们具有一致最小方差，并且在被估计的（协）方差所有可能参数值的整个范围内尽可能达到最大程度的无偏性。这是通过确定二次型矩阵元素的限制条件来实现的，这些限制条件是满足一致最小方差无偏估计的莱曼 - 谢费准则所必需的，然后通过最小二乘法原理求解由此产生的线性方程。本文的背景是任何混合线性模型，并且该方法不要求在多变量数据情况下观测值数量相等。文中给出了该方法的详细推导过程。讨论了该过程的完全通用性。提出了一种强制无偏性的修正方法。提供了一个具有三个方差分量模型的示例并讨论了结果。在杂项部分讨论了，除其他主题外，该方法如何用于比较其他（协）方差分量估计过程。最后一部分通过详细说明双变量模型所涉及的表达式，阐述了该方法如何处理多变量情况（即同时具有方差和协方差的模型）。
