Distribution of Parameter of Nonlinear Regression

Definition

Suppose a Nonlinear Regression Model $Y_i=f({\mathbf{X}}_i,\mathbf{\theta })+{ϵ}_i,i=1,2,\dots ,n$ If the error term ${ϵ}_i$ follows a Normal Distribution, LSE $\hat{\mathbf{\theta }}$ is the same as MLE.

For a large sample size, the distribution of the estimated parameter $\hat{\mathbf{\theta }}$ follows a Normal Distribution $\hat{\mathbf{\theta }}\stackrel{D}{\to }N(\mathbf{\theta },({\hat{\mathbf{J}}}^{⊺}\hat{\mathbf{J}}{)}^{-1}{\sigma }^2)$ where $\hat{\mathbf{J}}=\left(\frac{\partial f({\mathbf{X}}_i,\mathbf{\theta })}{\partial {\theta }_j}\right){|}_{\mathbf{\theta }=\hat{\mathbf{\theta }}}$ is the Jacobian Matrix of $f$ and ${\sigma }^2$ is the variance of the error term.

The variance of the error term can be estimated by $s^2=\frac{{\sum }_{i=1}^n(Y_i-{\hat{Y}}_i{)}^2}{n-p}=\frac{{\sum }_{i=1}^n(Y_i-f({\mathbf{X}}_i,\hat{\mathbf{\theta }}){)}^2}{n-p}$