We investigate the problem of compound estimation of normal means while accounting for the presence of side information. Leveraging the empirical Bayes framework, we develop a nonparametric integrative Tweedie (NIT) approach that incorporates structural knowledge encoded in multivariate auxiliary data to enhance the precision of compound estimation. Our approach employs convex optimization tools to estimate the gradient of the log-density directly, enabling the incorporation of structural constraints. We conduct theoretical analyses of the asymptotic risk of NIT and establish the rate at which NIT converges to the oracle estimator. As the dimension of the auxiliary data increases, we accurately quantify the improvements in estimation risk and the associated deterioration in convergence rate. The numerical performance of NIT is illustrated through the analysis of both simulated and real data, demonstrating its superiority over existing methods.

翻译：我们研究在存在辅助信息情况下正态均值复合估计问题。基于经验贝叶斯框架，我们提出一种非参数整合型特威迪（NIT）方法，该方法通过结合多变量辅助数据中编码的结构性知识来提高复合估计的精度。我们的方法采用凸优化工具直接估计对数密度的梯度，从而能够纳入结构性约束。我们对NIT的渐近风险进行理论分析，并确定NIT收敛于理想估计量的速率。随着辅助数据维度的增加，我们精确量化了估计风险的改善程度以及随之而来的收敛速度退化。通过模拟数据和真实数据的分析，我们展示了NIT方法的数值表现，证明了其相对于现有方法的优越性。