Estimation of High-Dimensional Normal Means through Inferential Models
The classical multivariate normal means problem remains conceptually unresolved. While shrinkage and empirical Bayes methods improve risk by imposing external geometric or hierarchical structure, they fail to explain how information is shared across independent coordinates for a fixed, unstructured mean vector. We address this gap using the prior-free Inferential Models framework. By formulating a generalized probability integral transform (GPIT) for independent, non-i.i.d~observations combined with a reweighted Anderson-Darling predictive random set, we leverage the global shape of ordered observations for valid, efficient inference. Crucially, the auxiliary structure of this formulation provides a novel explanation for Stein's paradox, demonstrating that the maximum likelihood estimator becomes structurally implausible for $n\geq 3$. To ensure scalability, we introduce an i.i.d. sampling-with-replacement surrogate that connects our exact fixed-mean formulation to overparameterized $g$-modeling. Furthermore, we develop a maximin criterion for combining plausibility contours. Under squared error loss, our estimators are competitive with state-of-the-art auto-modeling methods and outperform classical shrinkage and empirical Bayes methods.