New Multivariate-Distribution Estimators in view of Decision Theory

1.得到了如下结果：1）对于Stein损失函数，证明了矩阵参数的非保序估计是不可容许的.我们在二维情形下证明了这一点。2）我们发现了正交矩阵群上带状多项式的一个积分不等式是上述定理成立的一个充分条件。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性的如下结果：1）基于先前得到的渐近分布，我们导出了这些分布的高阶展开式; 2）利用这些展开式，我们导出了协方差矩阵的估计量是尾极小极大的必要条件。我们把这个结果应用到Stein和Haff估计中，发现它们都不是极小极大的.我们考虑了协方差矩阵等于给定矩阵的零假设的单侧检验。60年代已证明了关于特征值具有单调临界区域的不变检验的无偏性。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.