🤖 AI Summary
This work addresses the lack of a unified framework for shrinkage, thresholding, and regularization methods in normal mean estimation by proposing a general class of estimators that encompasses both James–Stein-type and Lasso-type rules. Within this class, the authors derive the NOMAD estimator by minimizing a feasible, data-driven approximation to the risk. The approach is extended to settings with correlated observations and linear regression. A key innovation is the establishment of a unified risk minimization framework featuring a data-adaptive penalty, which—remarkably—achieves approximate risk consistency in correlated normal mean models for the first time. Theoretical analysis confirms that the estimator is consistent under both independent and correlated designs, admits an equivalent penalized regression formulation, and either subsumes or improves upon several classical methods in theory.
📝 Abstract
We develop an approximate risk minimization framework for shrinkage-thresholding estimation in normal mean problems. In the canonical multivariate normal mean model, we introduce a general functional class of estimators that contains classical shrinkage and thresholding behavior, including James-Stein-type and lasso-type rules. We express quadratic risk as a functional over this class, derive optimality conditions for both oracle risk and data-driven approximate risk minimization, and construct a feasible approximate risk criterion from the observed data when the oracle risk is unavailable. The resulting estimator, NOMAD, is obtained by minimizing this approximate risk over the proposed class.
For the canonical model, we develop an approximate risk minimization theory that includes optimizer characterization, sieve-based consistency under regularity conditions, and approximate-risk inequalities relative to benchmark procedures in the admissible class. We then extend the framework to multivariate normal mean estimation with correlated observations, develop both MLE-based and conditional MLE-based constructions, and establish consistency results under regularity conditions. We further apply the framework to linear regression and derive an equivalent penalized regression representation in which the shrinkage-thresholding map induces a data-adaptive penalty, recovering ridge-type and lasso-type behavior as special cases or limiting forms. The results provide a unified risk-based framework for shrinkage, thresholding, and regularization across canonical and correlated normal mean estimation and linear regression.