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.

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