正态均值估计中收缩阈值规则下的近似风险最小化
Approximate Risk Minimization Over Shrinking-Thresholding Rules in Normal Mean Estimation
AI总结：
该研究为正态均值估计中的收缩阈值规则构建近似风险最小化框架，推导最优条件，得到估计器NOMAD，发展相关理论并扩展框架到相关观测及线性回归，提供了统一的基于风险的框架。
摘要：
我们为正态均值问题中的收缩阈值估计开发了一个近似风险最小化框架。在多元正态均值模型中，引入了包含经典收缩和阈值行为的估计器泛函类。将二次风险表示为该类上的泛函，推导了神谕风险和数据驱动近似风险最小化的最优条件。当神谕风险不可用时，从观测数据构建可行的近似风险准则。通过在所提出的类上最小化此近似风险得到估计器NOMAD。对于经典模型，发展了近似风险最小化理论。然后将框架扩展到相关观测的多元正态均值估计，建立了正则条件下的一致性结果。还将框架应用于线性回归，得到等效的惩罚回归表示。结果为收缩、阈值化和正则化提供了统一的基于风险的框架。
英文摘要：
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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