This paper offers a new approach for study the frequentist properties of the penalized MLE for general nonlinear regression models. The idea of the approach is to relax the nonlinear structural equation by introducing an auxiliary parameter for the regression response and replacing the structural equation with a penalty. This leads to a general semiparametric problem which is studied using the SLS approach from \cite{Sp2022}. We state sharp bounds on concentration and on the accuracy of the penalized MLE, Fisher and Wilks expansions, evaluate the risk of estimation over smoothness classes, and a number of further results. All the bounds are given in terms of effective dimension and do not involve the ambient dimension of the parameter space.

翻译：本文提出了一种新方法，用于研究一般非线性回归模型中惩罚极大似然估计的频率性质。该方法的核心思想是通过引入一个关于回归响应的辅助参数来宽松非线性结构方程，并用惩罚项替代该结构方程，从而得到一个一般的半参数问题，并采用文献\cite{Sp2022}中的SLS方法进行研究。本文给出了关于浓度与惩罚极大似然估计精度的尖锐边界、Fisher与Wilks展开、光滑性类上的估计风险评估，以及一系列进一步的结果。所有边界均以有效维度的形式给出，不涉及参数空间的环境维度。