On Exact and Efficient Inference for Many Normal Means

Inference about the unknown means $\theta=(\theta_1,...,\theta_n)' \in \mathbb{R}^n$ in the sampling model $X\sim N_n(\theta, I)$ from the observed $X$, known as the many-normal-means problem, has proven to be fundamental and yet challenging inasmuch as a satisfactory understanding remains elusive. To tackle exact and efficient inference about $\theta$, in this paper we propose innovative formulations of Inferential Models for two kinds of this problem: the {\it classic} kind given as is and the {\it empirical-Bayes} kind where $\theta$'s are further assumed to be an unobservable sample from an unknown non-parametric distribution $G(.)$. The formulation for the empirical-Bayes kind via numerical deconvolution allows for prior-free probabilistic inference with over-parameterization for the non-parametric model $G(.)$, whereas the formation for the first kind utilizes a latent random permutation and as a result provides a sound reasoning with uncertainty toward a deeper understanding. For uncertainty quantification with the more familiar frequentist inference framework, the method of maximum plausibility estimation is used for point estimation. Exact but conservative interval estimation is obtained based on plausibility, with a Monte Carlo based adaptive-adjustment approach to constructing shorter confidence intervals with targeted coverage. These methods are illustrated via simulation studies and a real-data example. The numerical results show that for interval estimation, adaptive intervals are satisfactory in both coverage and efficiency and that for point estimation, the proposed methods outperform the traditional James-Stein and Efron's g-modeling in terms of mean square error. The paper concludes with a few remarks, including future developments and extensions of the proposed methods.