Nonlinear Regression Model

Definition

Let $Y$ be a response and $\mathbf{X}=(X_1,X_2,\dots ,X_k{)}^{⊺}$ be $k$-dimensional explanatory variable. The non-linear regression model can be written as $Y_i=f({\mathbf{X}}_i,\mathbf{\theta })+{ϵ}_i,i=1,2,\dots ,n$ where $f({\mathbf{X}}_i,\mathbf{\theta })$ is a known regression function and $\mathbf{\theta }=({\theta }_1,{\theta }_2,\dots ,{\theta }_p{)}^{⊺}$ is a $p$-dimensional parameter vector.

The parameter vector $\mathbf{\theta }$ can not be obtained analytically. So it estimated numerically with such methods Newton–Raphson method, Gauss-Newton Method

Facts

If the function $f$ is linear, then the model is Multiple Linear Regression.