New Multivariate-Distribution Estimators in view of Decision Theory

1. 对于密度被展开为一系列带非负定矩阵参数的区域多项式的非负定随机矩阵，我们得到了正交等变估计量。1)对于Stein的损失函数，推测矩阵参数的非保序估计是不可接受的。我们在二维的情况下证明了这个。2)我们发现正交矩阵群上的分区多项式上的一个积分不等式是上述猜想成立的充分条件。通过与Takemura教授（东京大学）的共同研究，我们发现了Wishart矩阵的总体特征值无限分散时的渐近结果。1)基于之前得到的渐近分布，我们得到了渐近分布的高阶展开式。2)利用展开式导出了协方差矩阵的估计量为尾极小极大的必要条件。我们将这个结果应用到Stein和Haff估计中，发现它们都不是极小极大的。我们考虑对协方差矩阵等于给定矩阵的零假设进行单侧检验。关于特征值有单调临界域的不变检验的无偏性在60年代已经得到了证明。我们为这个结果找到了另一种证明。

1. We gained the following result on orthogonally equivariant estimators for a non-negative definite random matrix whose density is expanded as a series of zonal polynomials with a non-negative definite matrix parameter.1) With respect to Stein's loss function, non-order-preserving estimators for the matrix parameter are conjectured to be inadmissible. We proved this for 2-dimensional case.2) We found an integral inequality on zonal polynomials on the group of orthogonal matrices is a sufficient condition for the above conjecture.2. Through joint research with Professor Takemura (Tokyo University), we found the following result on an asymptotics when the population eigenvalues of Wishart matrix is infinitely dispersed.1) Based on the asymptotic distributions which were previously gained, we derived higher-order expansions for those distributions.2) We derived using the expansions a necessary condition for an estimator of the covariance matrix to be tail-minimax. We applied this result to Stein and Haff estimators and found neither of them is minimax.3. We considered one-sided test for the null hypothesis that the covariance matrix is equal to a given matrix. The unbiasedness for an invariant test which has a monotone critical region with respect to the eigenvalues was already proved in 60's. We found an alternative proof for the result.