Joint Confidence Region for Nonlinear Regression

Definition

Suppose a Nonlinear Regression Model $Y_i=f({\mathbf{X}}_i,\mathbf{\theta })+{ϵ}_i,i=1,2,\dots ,n$ The $100(1-\alpha )\%$ join confidence region for $\mathbf{\theta }$ is obtained as $\{\mathbf{\theta }|(\mathbf{\theta }-\hat{\mathbf{\theta }}{)}^{⊺}{\hat{\mathbf{J}}}^{⊺}\hat{\mathbf{J}}(\mathbf{\theta }-\hat{\mathbf{\theta }})\le p{s}^2{F}_{\alpha }(p,n-p)\}$ where $s^2=\frac{{\sum }_{i=1}^n(Y_i-f({\mathbf{X}}_i,\hat{\mathbf{\theta }}){)}^2}{n-p}$, $p-1$ is the number of explanatory variables, and $\hat{\mathbf{J}}=\left(\frac{\partial f({\mathbf{X}}_i,\mathbf{\theta })}{\partial {\theta }_j}\right){|}_{\mathbf{\theta }=\hat{\mathbf{\theta }}}$ is the Jacobian Matrix of $f$.