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Approximate Risk Minimization Over Shrinking-Thresholding Rules in Normal Mean Estimation

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.

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