Which estimation technique is used when the relationship between variables and parameters is nonlinear and you fit by minimizing the squared error in the predicted observations?

Which estimation technique is used when the relationship between variables and parameters is nonlinear and you fit by minimizing the squared error in the predicted observations?

• A. Nonlinear Least Squares (NLS)
• B. Maximum Likelihood Estimation (MLE)
• C. Ordinary Least Squares (OLS)
• D. Residual Sum of Squares (RSS)

Answer: (A)

Nonlinear relationships between variables and parameters call for an estimation approach that directly optimizes the parameter values in a nonlinear model. Nonlinear Least Squares fits by finding the parameter values that minimize the sum of squared differences between the observed data and the model’s predictions. Because the predicted observations are a nonlinear function of the parameters, there isn’t a simple closed-form solution like in linear regression; instead, you solve it with iterative optimization (for example, using Gauss-Newton or Levenberg–Marquardt methods) to minimize the squared residuals. This is distinct from ordinary least squares, which assumes the relationship is linear in the parameters. Maximum Likelihood Estimation is a broader framework that becomes equivalent to least squares only under specific error assumptions (like Gaussian errors) but isn’t defined by minimizing squared errors alone. Residual Sum of Squares is the objective being minimized in least squares methods, not the estimation technique itself. So, the method that best fits a nonlinear relationship by minimizing the squared error in predicted observations is Nonlinear Least Squares.

Nonlinear relationships between variables and parameters call for an estimation approach that directly optimizes the parameter values in a nonlinear model. Nonlinear Least Squares fits by finding the parameter values that minimize the sum of squared differences between the observed data and the model’s predictions. Because the predicted observations are a nonlinear function of the parameters, there isn’t a simple closed-form solution like in linear regression; instead, you solve it with iterative optimization (for example, using Gauss-Newton or Levenberg–Marquardt methods) to minimize the squared residuals.

This is distinct from ordinary least squares, which assumes the relationship is linear in the parameters. Maximum Likelihood Estimation is a broader framework that becomes equivalent to least squares only under specific error assumptions (like Gaussian errors) but isn’t defined by minimizing squared errors alone. Residual Sum of Squares is the objective being minimized in least squares methods, not the estimation technique itself.

So, the method that best fits a nonlinear relationship by minimizing the squared error in predicted observations is Nonlinear Least Squares.