Maximum likelihood estimator-like defining \(\rho = -\log{f}\) and solving $$min_{\hat{\beta}}\sum{\rho(y_i-x_i\hat{\beta})}$$ \(\psi = -\rho'\) is the influence curve and creates the system of \(k+1\) estimating equations for the coefficients: $$\sum_{i=1}^n{\psi(y_i-x_i'\hat{\beta})x_i'}=0$$ The weight functions are defined \(w(e)=\frac{\psi(e)}{e}\) and \(w_i=w(e_i)\).
Solving is a weighted least-squares problem, minimizing \(\sum{w_i^2e_i^2}\).
Weighted Least Squares: \(\beta = [X'WX]^{-1}X'WY\).
Scale \(s_n(u_1, ..., u_n)\) is defined by the value of \(s\) satisfying $$\frac{1}{n-p}\sum_{i=1}^n{\chi(\frac{y_i-x_i\hat{\beta}}{s})}=E_\phi(\rho)$$