Minimax Estimation of a Normal Mean Vector When the Covariance Matrix is Unknown

研究问题: 在维度至少为三、协方差未知的多元正态模型中，能否利用独立的 Wishart 矩阵构造收缩估计，在指定二次损失下改善风险？
方法与思路: 作者研究由独立 Wishart 矩阵、已知权重矩阵及满足单调条件的函数构成的收缩估计类，寻找保证极小极大性质的参数区间。
材料陈述的贡献: 摘要称，在给定参数区间内，该估计类具有极小极大性质并优于通常的观测向量估计；自由度与维度接近时区间可能退化。
阅读关注点: 核对损失的归一化、独立性、函数约束及自由度条件，查阅原文确认符号定义、参数区间和风险支配在边界处的含义。
阅读边界: 仅据 OpenAlex 索引摘要，稿件版本未注明；公式中的 n* 未定义，须核对原文；风险优势限于指定损失和条件，退化区间未必给出严格改进。

GPT-6.1 Sol · 高 · 2026-10-07

原文摘要

Let $X$ be an observation from a $p$ -variate normal distribution $(p \geqslant 3)$ with mean vector $\theta$ and unknown positive definite covariance matrix $\sum$ . We wish to estimate $\theta$ under the quadratic loss $L(\delta; \theta, \sum) = \lbrack\mathrm{tr}(Q\sum)\rbrack^{-1}(\delta - \theta)'Q(\delta - \theta)$ , where $Q$ is a known positive definite matrix. Estimators of the following form are considered: $\delta_{k, h}(X, W) = \lbrack I - kh(X'W^{-1}X)\lambda_1(QW/n^\ast)Q^{-1}W^{-1}\rbrack X,$ where $W : p \times p$ is observed independently of $X$ and has a Wishart distribution with $n$ degrees of freedom and parameter $\sum, \lambda_1(A)$ denotes the minimum characteristic root of $A$ , and $h(t): \lbrack 0, \infty) \rightarrow \lbrack 0, \infty)$ is absolutely continuous with respect to Lebesgue measure, is nonincreasing, and satisfies the additional requirements that $th(t)$ is nondecreasing and $\sup_{t \geqslant 0}th(t) = 1$ . With $h(t) = t^{-1}$ , the class $\delta_{k, h}$ specializes to that considered by Berger, Bock, Brown, Casella and Gleser (1977). For the more general class considered in the present paper, it is shown that there is an interval $\lbrack 0, k_{n, p}\rbrack$ of values of $k$ (which may be degenerate for small values of $n - p)$ for which $\delta_{k, h}$ is minimax and dominates the usual estimator $\delta_0 \equiv X$ in risk.Let $X$ be an observation from a $p$-variate normal distribution $(p \geqslant 3)$ with mean vector $\theta$ and unknown positive definite covariance matrix $\sum$. We wish to estimate $\theta$ under the quadratic loss $L(\delta; \theta, \sum) = \lbrack\mathrm{tr}(Q\sum)\rbrack^{-1}(\delta - \theta)'Q(\delta - \theta)$, where $Q$ is a known positive definite matrix. Estimators of the following form are considered: $\delta_{k, h}(X, W) = \lbrack I - kh(X'W^{-1}X)\lambda_1(QW/n^\ast)Q^{-1}W^{-1}\rbrack X,$ where $W : p \times p$ is observed independently of $X$ and has a Wishart distribution with $n$ degrees of freedom and parameter $\sum, \lambda_1(A)$ denotes the minimum characteristic root of $A$, and $h(t): \lbrack 0, \infty) \rightarrow \lbrack 0, \infty)$ is absolutely continuous with respect to Lebesgue measure, is nonincreasing, and satisfies the additional requirements that $th(t)$ is nondecreasing and $\sup_{t \geqslant 0}th(t) = 1$. With $h(t) = t^{-1}$, the class $\delta_{k, h}$ specializes to that considered by Berger, Bock, Brown, Casella and Gleser (1977). For the more general class considered in the present paper, it is shown that there is an interval $\lbrack 0, k_{n, p}\rbrack$ of values of $k$ (which may be degenerate for small values of $n - p)$ for which $\delta_{k, h}$ is minimax and dominates the usual estimator $\delta_0 \equiv X$ in risk.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）

笔记与分组

分类依据与纠错

Minimax Estimation of a Normal Mean Vector When the Covariance Matrix is Unknown

代码、数据与补充材料

协方差矩阵未知时正态均值向量的极小极大估计

原文摘要

同方向的基础文献

同方向的基础文献