Introduction

Abstract

This monograph provides an introduction to a class of linear fitting procedures that employ the sum of the absolute residuals (or $L_{1}$-norm), the minimax absolute residual (or $L_{\infty }$-norm) and the median squared residual as optimality criteria in the context of the standard linear statistical model. The least absolute residuals procedure was proposed by Boscovich in 1757 and again in 1760 and discussed by Laplace, Gauss and Edgeworth; the least squares procedure was probably used by Gauss in 1794 or 1795 but first proposed in print by Legendre in 1805 before being discussed by Gauss, Laplace and many other leading scientists; finally, the minimax absolute residual procedure was proposed by Laplace in 1786, 1793 and 1799 before being discussed by Cauchy, Fourier, Chebyshev and others. The least squares and least absolute residuals procedures are widely used in statistical applications but the minimax procedure had received little support in this area until a variant, the least median of squares procedure, was proposed by Rousseeuw in 1984. Almost by definition, this last procedure is more robust to the presence of outlying observations than are the other two fitting procedures.