未知方差下回归系数常用估计的可容许性

假设一个”维随机向量Y服从正态分布N(β，σ^2In)。在二次损失下，当n≥3和σ^2已知时，Stein在1956年指出Y不是β的容许估计，这是统计判决理论中一个著名的结果，成平在1982年对Stein结果给出了一个有趣的补充，他证明了当σ^2未知时，Y是β的容许估计。这篇文章是成平结果的一般化，即在一个宽广的分布类中，证明了当方差未知时，回归系数最小二乘估计是容许的。这表明当方差未知时，回归系数最小二乘估计是一个适合的估计。

Admissibility of the usual estimator for a regression coeffcient under unknown variance

Suppose an n-dimension random vector Y is distributed to the normal distribution N(β,σ2In). When σ2 is known and n≥3, Stein[1] pointed out that Y is an inadmissible estimator of β under the quadratic loss. This is a well known result in statistical decision theory. Cheng[2] gave an interesting supplement of Stein's result. He proved that Y is admissible when σ2 is unknown. This paper is a generalization of the result in [2]. We prove that the least squares estimator of the regression coeffcient is admissible for a wide class of distributions when the variance σ2 is unknown. This shows that the usual estimator of the regression coeffcient is available when σ2 is unknown.