L1-norm and L[infinity symbol]-norm estimation : an introduction to the least absolute residuals, the minimax absolute residual and related fitting procedures

Introduction -- Point Fitting Problems in One and Two Dimensions -- The Hyperplane Fitting Problem in Two or More Dimensions -- Linear Programming Computations -- Statistical Theory -- The Least Median of Squared Residuals Procedure -- Mechanical Representations Cover......Page 1 L1-Norm andL?-Norm Estimation......Page 3 Contents......Page 5 7 Mechanical Representations......Page 7 References......Page 8 2.1 Point Fitting Problems in One Dimension......Page 10 2.2 Point Fitting Problems in Two Dimensions......Page 12 2.3 Truncated Point Fitting Problems in Two Dimensions......Page 13 References......Page 14 3.1 Line Fitting Problems in Two Dimensions......Page 15 3.2 Hyperplane Fitting Problems in Higher Dimensions......Page 16 3.3 Matrix Representation of the Problem......Page 17 3.4 The Weighted l1-Norm and l2-Norm Fitting Problems......Page 18 3.5 Relation to the L2-Norm Fitting Problem......Page 22 References......Page 23 4.1 Regular and Projective Geometry Solutions of the L1-Norm Problem......Page 24 4.3 Linear Programming Computations......Page 26 4.4 A Hypothetical General Procedure......Page 27 4.5 Non-Uniqueness of the Solution......Page 28 References......Page 30 5.1 Linear Statistical Model......Page 33 5.2 Maximum Likelihood Estimation......Page 34 5.3 Asymptotic Theory......Page 35 5.4 Pseudo-Unbiased Weighted L1-Norm Procedures......Page 36 5.5 Pseudo-Unbiased Weighted hxiuhd-Norm Procedures......Page 37 References......Page 38 6.1 Robustness to Outliers......Page 39 6.3 Least Median of Squares Computations......Page 40 6.4 Minimum Volume Ellipsoid......Page 41 References......Page 42 7.1 Introduction......Page 44 7.2 Fitting a Point to a Set of Points in the Plane of Observations: Fermat's Problem......Page 45 7.3 Fitting a Line to a Set of Points in the Plane of Observations: Boscovich's Problem......Page 47 7.4 Fitting a Point to a Set of Lines in the Plane of Parameters......Page 49 7.5 Fitting a Point to a Set of Lines in the Plane of Observations: Oja's Bivariate Median......Page 51 7.6 L2-Norm Mechanical Models......Page 53 7.7 LinftyNorm and LMS Mechanical Models......Page 55 References......Page 56 Author Index......Page 58 This monograph is concerned with the fitting of linear relationships in the context of the linear statistical model. As alternatives to the familiar least squared residuals procedure, it investigates the relationships between the least absolute residuals, the minimax absolute residual and the least median of squared residuals procedures. It is intended for graduate students and research workers in statistics with some command of matrix analysis and linear programming techniques.​ Introduction Point Fitting Problems in One- and Two-dimensions The Hyperplane Fitting Problem in Two or More Dimensions Linear Programming Computations Statistical Theory The Least Median of Squared Residuals Procedure Mechanical Representations References Index of Names. ​ .