Many statistics problems are formulated within an estimating equation framework instead of a minimization framework. However, the regularized estimating equations (REE) have been much less extensively studies than regularized minimization problems. In this paper, we study an improved regularized estimating equation formulation and explore its subsequent equivalences in terms of (1) fixed-point problem specified via the proximal operator of the corresponding regularizer, and (2) generalized variational inequality problems. Such equivalences hold under general conditions and accommodate nonconvex regularizers. Moreover, these equivalences open up new possibilities in theoretical analysis and computational algorithms when studying the REE.

翻译：许多统计学问题是在估计方程框架而非优化框架下构建的。然而，相较于正则化最小化问题，正则化估计方程的深入研究仍显不足。本文研究了一种改进的正则化估计方程公式，并探讨了其与以下两种问题的等价性：（1）通过对应正则化项邻近算子定义的不动点问题；（2）广义变分不等式问题。在一般性条件下，该等价关系成立且适用于非凸正则化项。此外，这些等价性为研究正则化估计方程的理论分析和计算算法开辟了新的可能性。