New Multivariate-Distribution Estimators in view of Decision Theory

1)非负定随机矩阵的密度展开为一系列带非负定矩阵参数的带状多项式。1)对于Stein损失函数，关于矩阵参数的非保序估计被猜想为不可接受的。我们在2维情形下证明了这一点。2)我们发现关于正交矩阵群上的带状多项式的一个积分不等式是上述猜想成立的一个充分条件。通过与Takemura教授(东京大学)的合作研究，我们得到了当Wishart矩阵的总体特征值是无穷离散时的一个渐近性结果。1)基于已有的渐近分布，我们得到了这些分布的高阶展开式。2)利用这些展开式，我们得到了协方差阵的估计是尾极小极大的一个必要条件。我们将这一结果应用于Stein和Haff估计，发现它们都不是极小极大。我们考虑了协方差矩阵等于给定矩阵的零假设的单边检验。关于特征值具有单调临界区的不变检验，60‘S已经证明了它的无偏性，我们给出了另一种证明。

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.