Estimating equations arise in a wide range of statistical applications, including longitudinal and clustered data analysis, survival analysis, econometrics, and semiparametric inference. In high-dimensional settings, adding sparsity-inducing regularization often leads to computational challenges that are not fully addressed by standard penalized optimization routines. These challenges are closely tied to the structural form of the underlying estimating problem: mainly, the estimating function needs not be the gradient of a scalar objective and may involve asymmetric Jacobians, overidentification, nonsmoothness, nonconvexity, or nested optimization. This article first reviews the application areas of estimating equations, and then the computational methods for regularized estimating equations by organizing them into four broad formulations: minimization-type, Dantzig-type, regularization-type, and fixed-point-type approaches. We discuss the main numerical strategies associated with each formulation, including penalized optimization, constrained linear programming, iterative root-solving, and proximal fixed-point iteration. We also highlight the connection between regularized estimating equations and fixed-point problems, which provides a unified computational perspective for analyzing and solving regularized estimating equations.

翻译：估计方程广泛出现在各类统计应用中，包括纵向和聚类数据分析、生存分析、计量经济学以及半参数推断。在高维场景下，加入稀疏性诱导正则化往往会给计算带来挑战，而标准的惩罚优化方法无法完全解决这些问题。这些挑战与估计问题的结构形式密切相关：主要在于估计函数未必是标量目标函数的梯度，且可能涉及非对称雅可比矩阵、过度识别、非光滑性、非凸性或嵌套优化。本文首先回顾了估计方程的应用领域，随后将正则化估计方程的计算方法归纳为四大类：最小化型、Dantzig型、正则化型和不动点型方法。我们讨论了每类方法对应的主要数值策略，包括惩罚优化、约束线性规划、迭代求根以及近端不动点迭代。同时，本文揭示了正则化估计方程与不动点问题之间的内在联系，这为分析和求解正则化估计方程提供了统一的计算视角。