论文标题
另一个看许多正常均值的问题
Another Look at the Problem of Many-Normal-Means
论文作者
论文摘要
在多变量普通模型$ x \ sim n_n（θ，i）$中推断均值的平均值向量$θ=（θ_1，...，...，θ_n）'\ in \ Mathbb {r}^n $和观察到的data $ x =（x_1，...，...，...，x_n）\ in { （MNMS）。本文在推论模型（IMS）的框架内解决了两种基本的MNM。第一种被称为{\ it Classic}种类的一种。第二种，称为{\ it经验贝叶斯}，假设个人表示$θ_i$的$是从未知的分布$ g（。）$中独立绘制的{\ it先验{\ it先验}。经验贝叶斯类型的IM配方利用了数值反卷积，从而实现了$ g（。）$的过度参数化的先前概率推断。另一方面，经典类型的IM表述利用了潜在的随机排列，为不确定性和更深入的理解提供了一种新颖的方法。对于熟悉的频繁推理框架中的不确定性定量，最大值的IM方法用于点估计。使用基于蒙特卡洛的自适应调整方法来构建具有目标覆盖范围的较短置信区间，可以根据合理性获得保守的间隔估计。这些方法是通过仿真研究和真实数据示例来证明的。数值结果表明，在均方误差方面，提出的点估计方法优于传统的詹姆斯·斯坦（James-Stein）和埃夫隆（Efron）的$ g $模型，并且自适应间隔在覆盖范围和效率方面都令人满意。本文以建议未来的发展和拟议方法的扩展为结论。
Inferring the means in the multivariate normal model $X \sim N_n(θ, I)$ with unknown mean vector $θ=(θ_1,...,θ_n)' \in \mathbb{R}^n$ and observed data $X=(X_1,...,X_n)'\in {\mathbb R}^n$ is a challenging task, known as the problem of many normal means (MNMs). This paper tackles two fundamental kinds of MNMs within the framework of Inferential Models (IMs). The first kind, referred to as the {\it classic} kind, is presented as is. The second kind, referred to as the {\it empirical Bayes} kind, assumes that the individual means $θ_i$'s are drawn independently {\it a priori} from an unknown distribution $G(.)$. The IM formulation for the empirical Bayes kind utilizes numerical deconvolution, enabling prior-free probabilistic inference with over-parameterization for $G(.)$. The IM formulation for the classic kind, on the other hand, utilizes a latent random permutation, providing a novel approach for reasoning with uncertainty and deeper understanding. For uncertainty quantification within the familiar frequentist inference framework, the IM method of maximum plausibility is used for point estimation. Conservative interval estimation is obtained based on plausibility, using a Monte Carlo-based adaptive adjustment approach to construct shorter confidence intervals with targeted coverage. These methods are demonstrated through simulation studies and a real-data example. The numerical results show that the proposed methods for point estimation outperform traditional James-Stein and Efron's $g$-modeling in terms of mean square error, and the adaptive intervals are satisfactory in both coverage and efficiency. The paper concludes with suggestions for future developments and extensions of the proposed methods.
