Estimation of High-Dimensional Normal Means through Inferential Models

The estimation of the multivariate normal mean is a fundamental problem, highlighted by the inadmissibility of the MLE for $n\ge 3$ under quadratic loss. While shrinkage and empirical Bayes methods leverage joint structure through geometric reasoning or hierarchical modeling, this paper proposes a class of point estimators derived from the prior-free framework of inferential models. We develop a generalized probability integral transform for independent, non-i.i.d observations, creating a bijective mapping from the sample to an ordered-uniform reference distribution. By combining this bijection with an ordered-uniform predictive random set based on a reweighted Anderson-Darling statistic, we ensure valid and efficient inference that captures the global shape structure revealed by the ordered observations. We further introduce a maximin (bottleneck) criterion for combining multiple plausibility contours. To ensure computability, we develop a sampling-with-replacement surrogate that connects the exact formulation to over-parameterized (g)-modeling. Our approach provides a structural explanation of Stein’s paradox, showing that the MLE corresponds to a zero-density point of the joint auxiliary distribution, revealing its implausibility from an auxiliary perspective. Numerical studies show that our estimators are competitive with state-of-the-art auto-modeling methods and outperform classical shrinkage and empirical Bayes methods.