Research on Shrinkage Estimator

关于二元正态协方差矩阵的广义Bayes估计的精确风险函数，由Yang和Berger(1994，Ann.统计学家。)是派生出来的。并与Takemura的Stein极小极大估计、收缩估计、Haff估计和正交等变估计进行了比较。发现广义贝叶斯估计在潜根之比远离1的某一区域内并不支配极大似然估计，而在1附近是非常好的.给出了经典的二元线性假设的似然比、最大潜根、Hoteling迹和Pillai局部最好不变检验四个准则的精确幂的计算公式，并给出了幂曲面的图形.研究发现，在零假设附近的Hotling检验、似然比检验和远离零假设的逆不等检验的幂的排序并不一定成立，但在大样本情况下仍然成立.给出了循环有序正态均值的广义贝叶斯检验，它对于给定的非中心性具有几乎相同的最小功效，而比似然比检验具有更大的最大功效.发现非信息先验应取在由备选假设形成的多面体凸锥的最远角矢量上。结果表明，最远的角矢量是由两个成对的角矢量组成的，每个角矢量与圆锥体中心成等角度相交。

The exact risk function of the generalized Bayes estimator for bivariate normalcovariance matrix against reference prior due to Yang and Berger(1994, Ann. Statist.) is derived. It is compared with those of the Stein's minimax estimator, shrinkage estimator, Haff's estimator and orthogonally equivariant estimator by Takemura. It is found that the generalized Bayes estimator does not dominate maximum likelihood estimator at some region of the ratio of the latent root far from one and that it is very good near one.The formula for calculating the exact power of classically famous four criteria ; likelihood ratio, maximum latent root, Hotelling's trace and Pillai's locally best invariant test for bivariate linear hypotheses are derived and the graphs of the power surfaces are demonstrated to see the global behavior. It is found that the ordering of the power : Hotelling's test<Likelihood ratio test<Pillai's test at the neighborhood of the null hypotheses and the reversed inequalities at far from the null hypotheses do not necessarily hold, though they do hold for large sample sizes.The generalized Bayes test for loop-ordered normal means is derived which has almost the same minimum power for given non-centrality and has much larger maximum power than likelihood ratio test. It is found that the non-informative prior should be taken at the most distant corner vectors of the polyhedral convex cone formed by the alternative hypotheses. It is shown that the most distant corner vectors are made from pairs of two, each of which intersect the center of the cone with equal angle.