Gini Mean Difference

Alternatively, we can minimize the Gini Mean Difference of the $f_i(\theta).$ Now $$\begin{multline*} \sum_{i=1}^n\sum_{j=1}^n\mid f_i(\theta)-f_j(\theta)\mid \leq\\ \sum_{i=1}^n\sum_{j=1}^n\frac{1}{\mid f_i(\xi)-f_j(\xi)\mid} (f_i(\theta)-f_j(\theta))^2 + \text{terms}, \end{multline*}$$ which can be rewritten as $$\cdots=\sum_{i=1}^n\sum_{j=1}^n w_{ij}(\xi)f_i(\theta)f_j(\theta)+\text{terms},$$ minimization of which is a weighted least squares problem.