Nonlinear regression analysis is a popular and important tool for scientists and engineers. In this article, we introduce theories and methods of nonlinear regression and its statistical inferences using the frequentist and Bayesian statistical modeling and computation. Least squares with the Gauss-Newton method is the most widely used approach to parameters estimation. Under the assumption of normally distributed errors, maximum likelihood estimation is equivalent to least squares estimation. The Wald confidence regions for parameters in a nonlinear regression model are affected by the curvatures in the mean function. Furthermore, we introduce the Newton-Raphson method and the generalized least squares method to deal with variance heterogeneity. Examples of simulation data analysis are provided to illustrate important properties of confidence regions and the statistical inferences using the nonlinear least squares estimation and Bayesian inference.

翻译：非线性回归分析是科学家与工程师广泛使用的重要工具。本文介绍了基于频率学派和贝叶斯统计建模与计算的非线性回归理论、方法及其统计推断。基于高斯-牛顿法的最小二乘估计是最常用的参数估计方法。在误差服从正态分布的假设下，极大似然估计等价于最小二乘估计。非线性回归模型中参数的Wald置信区域受均值函数曲率的影响。此外，本文介绍了处理方差异质性的牛顿-拉夫逊法与广义最小二乘法。通过模拟数据分析示例，展示了置信区域的重要性质以及基于非线性最小二乘估计和贝叶斯推断的统计推断方法。