熵损失下协方差矩阵最佳仿射同变估计的改进(英)

Improvement on the Best Affine Equivariant Estimation of the Covariance Matrix under the Entropy Loss

设Ｘ１；…，Ｘｎ（ｎ＞ｐ）是来自多元正态分布Ｎｐ（μ，∑）的一个样本，其中μ∈Ｒ~ｐ，∑＞０均未知．本文在熵损失 Ｌ（ｓｕｍ ｆｒｏｍ ｔｏ ～，∑）＝ｔｒ（∑~－１，ｓｕｍ ｆｒｏｍ ｔｏ ～）－ｌｏｇ｜∑~－１ｓｕｍ ｆｒｏｍ ｔｏ～｜－ｐ下证明了协方差矩阵∑的最佳仿射同变估计是不容许的，且给出了其改进估计．

Let X1,…, X_n (n > p) be a random sample from multivariate normal distribution N_p(μ, ∑), where μ ∈RP and ∑ is a positive definite matrix, both μ and ∑ being unknown. In this paper it is shown for the entropy loss L(sum from to ^,∑ ) = tr(∑^-1 sum from to^) - log|Σ^-1 sum from to ^|-p the best affine equivariant estimator of the covariance matrix ∑ is inadmissible and an improved estimator is explicitly constructed.