随机变量二次型的协方差在混合效应模型中的应用

本文提出方差分量ANOVA估计的一种改进方法, 证明了对于一般的方差分量模型, 只要方差分量的ANOVA估计存在就可以通过此方法给出其改进形式, 并且在均方误差意义下优于ANOVA估计. 特别地, 对于单向分类随机效应模型, Kelly和Mathew^[1]对ANOVA估计的改进就是我们提出的改进方法的特殊形式, 这也给出了此类改进估计在均方误差意义下优于ANOVA估计的另一种合理的解释. 同时, 本文又将此思想应用到对谱分解估计的改进上. 本文应用协方差的简单性质证明了对带有一个随机效应的方差分量模型, 当随机效应的协方差阵只有一个非零特征值时, 随机效应方差分量谱分解估计在均方误差意义下总是优于ANOVA估计. 本文最后将第三节的结论推广到广义谱分解估计下, 同时给出广义谱分解估计待定系数的一个合理的取值.

The Covariances of Quadratic Forms of Random Variables and Their Applications in Linear Mixed Models

An improved ANOVA estimator is obtained in this paper. For the linear mixed model, the improved estimator can be obtained by this idea and dominates ANOVA estimator. For random effects model of one-way classification, the estimator considered by Kelly and Mathew^[1] is a special case. This idea is also used to improve spectral decomposition estimator. This paper applys a simple property of variance-covariance to prove that the spectral decomposition estimator uniformly dominates ANOVA estimator for random effects model with one variance component when the covariance matrix of random effects only has one eigenvalue. Finally, this result is extended and we obtain a better feasible generalized spectral decomposition estimator.