Consistency of the least weighted squares under heteroscedasticity

A robust version of the Ordinary Least Squares accommodating the idea of weighting the order statistics of the squared residuals (rather than directly the squares of residuals) is recalled and its properties are studied. The existence of solution of the corresponding extremal problem and the consistency under heteroscedasticity is proved. 1. BASIC FRAMEWORK AND WEIGHTING THE ORDER STATISTICS Let N denote the set of all positive integers, R the real line and R p the p-dimensional Euclidean space. All vectors will be assumed to be the column ones and throughout the paper, we assume that all r.v.'s are defined on a basic probability space (Ω, A, P). For a sequence of (p + 1)-dimensional random variables {(X