Nonlinear Lp-Norm Estimation

Description

Complete with valuable FORTRAN programs that help solve nondifferentiable nonlinear Lp and Linf-norm estimation problems, this important reference/text extensively delineates ahistory of Lp-norm estimation. It examines the nonlinear Lp-norm estimation problem that is a viable alternative to least squares estimation problems where the underlying error distribution is nonnormal, i.e., non-Gaussian. "Nonlinear Lp-Norm Estimation" addresses both computational and statistical aspects of Lp-norm estimation problems to bridge the gap between these two fields ... contains 70 useful illustrations ... discusses linear Lp-norm as well as nonlinear L1, Linf, and Lp-norm estimation problems ... provides all appropriate computational algorithms and FORTRAN listings for nonlinear Lp and Linf-norm estimation problems ... guides readers with clear end of-chapter notes on related topics and outstanding research publications ... contains numerical examples plus several practical problems ... and shows how the data can prescribe various applications of Lp-norm alternatives. "Nonlinear Lp-Norm Estimation" is an indispensable reference for statisticians, operations researchers, numerical analysts, applied mathematicians, biometricians, and computer scientists, as well as a text for graduate students in statistics or computer science. Cover Half Title Title Page Copyright Page PREFACE Table of Contents 1: Lp-NORM ESTIMATION IN LINEAR REGRESSION 1.1 The history of curve fitting problems 1.2 The linear Lp-norm estimation problem 1.2.1 Formulation 1.2.2 Algorithms 1.2.3 L2-estimation 1.2.4 L1-estimation 1.2.5 L∞-estimation 1.3 The choice of p 1.4 Statistical Properties of Linear Lp-norm Estimators 1.5 Confidence intervals for β 1.5.1 Case 1 < p < ∞ 1.5.2 Case p = 1 1.6 Example Conclusion Appendix 1A: Gauss-Markoff Theorem Additional notes Bibliography: Chapter 1 2: THE NONLINEAR L1-NORM ESTIMATION PROBLEM 2.1 The nonlinear L1-norm estimation problem 2.2 Optimality conditions for L1-norm estimation problems 2.2.1 Examples 2.3 Algorithms for solving the nonlinear L1- norm estimation problem 2.3.1 Differentiable unconstrained minimization 2.3.2 Type I Algorithms 2.3.2.1 The Anderson-Osborne-Watson algorithm 2.3.2.2 The Anderson-Osborne-Levenberg-Marquardt algorithm 2.3.2.3 The McLean and Watson algorithms 2.3.3 Type II Algorithms 2.3.3.1 The Murray and Overton algorithm 2.3.3.2 The Bartels and Conn algorithm 2.3.4 Type III methods 2.3.4.1 The Hald and Madsen algorithm 2.3.5 Type IV Algorithms 2.3.5.1 The El-Attar-Vidyasagar-Dutta algorithm 2.3.5.2 The Tishler and Zang algorithm Conclusion Appendix 2A: Local and global minima Appendix 2B: One-dimensional line search algorithms Appendix 2C: The BFGS approach in unconstrained minimization Appendix 2D: Levenberg-Marquardt approach in nonlinear estimation Appendix 2E: Rates of convergence Appendix 2F: Linear Algebra Appendix 2G: Extrapolation procedures to enhance convergence Appendix 2H: FORTRAN Programs Additional notes Bibliography: Chapter 2 3: THE NONLINEAR L∞-NORM ESTIMATION PROBLEM 3.1 The nonlinear L∞-norm estimation problem 3.2 Loo-norm optimality conditions 3.3 The nonlinear minimax problem 3.4 Examples 3.5 Algorithms for solving nonlinear Loo-norm estimation problems 3.5.1 Type I Algorithms 3.5.1.1 The Anderson-Osborne-Watson algorithm 3.5.1.2 The Madsen algorithm 3.5.1.3 The Anderson-Osborne-Levenberg-Marquardt algorithm 3.5.2 Type II Algorithms 3.5.2.1 The Murray and Overton algorithm 3.5.2.2 The Han algorithm 3.5.3 Type III Algorithms 3.5.3.1 The Watson algorithm 3.5.3.2 The Hald and Madsen algorithm 3.5.4 Type IV algorithms 3.5.4.1 The Charalambous acceleration algorithm 3.5.4.2 The Zang algorithm Appendix 3A: FORTRAN Programs Additional notes Bibliography: Chapter 3 4: THE NONLINEAR Lp-NORM ESTIMATION PROBLEM 4.1 The nonlinear Lp-norm estimation problem (1 < p < ∞) 4.2 Optimality conditions for the nonlinear Lp-norm problem 4.3 Algorithms for nonlinear Lp-norm estimation problems 4.3.1 Watson’s algorithm 4.3.2 A first-order gradient algorithm 4.3.2.1 Examples 4.3.3 The mixture method for large residual and ill-conditioned problems 4.3.3.1 Examples Conclusion Appendix 4A: Cholesky decomposition of symmetric matrices Appendix 4B: The singular value decomposition (SVD) of a matrix Appendix 4C: Fletcher’s line search algorithm Additional notes Bibliography: Chapter 4 5: STATISTICAL ASPECTS OF Lp-NORM ESTIMATORS 5.1 Nonlinear least squares 5.1.1 Statistical inference 5.1.1.1 Confidence intervals and joint confidence regions 5.1.1.2 Hypothesis testing 5.1.1.3 Bias 5.2 Nonlinear L1-norm estimation 5.2.1 Statistical inference 5.2.1.1 Confidence intervals 5.2.1.2 Hypothesis testing 5.2.1.3 Bias and consistency 5.3 Nonlinear Lp-norm estimation 5.3.1 Asymptotic distribution of Lp-norm estimators (additive errors) 5.3.2 Statistical inference 5.3.2.1 Confidence intervals 5.3.2.2 Hypothesis testing 5.3.2.3 Bias 5.4 The choice of the exponent p 5.4.1 The simulation study 5.4.2 Simulation results and selection of p 5.4.3 The efficiency of Lp-norm estimators for varying values of p 5.4.4 The relationship between p and the sample kurtosis 5.5 The asymptotic distribution of p 5.6 The adaptive algorithm for Lp-norm estimation 5.7 Critique of the model and regression diagnostics 5.7.1 Regression diagnostics (nonlinear least squares) 5.7.2 Regression diagnostics (nonlinear Lp-norm case) Conclusion Appendix 5A: Random deviate generation Appendix 5B: Tables Appendix 5C: ω2p for symmetric error distributions Appendix 5D: Var(pˆ) for symmetric error distributions Appendix 5E: The Cramér-von Mises goodness-of-fit test Additional notes Bibliography: Chapter 5 6: APPLICATION OF Lp-NORM ESTIMATION 6.1 Compartmental models, bioavailability studies in Pharmacology 6.1.1 Mathematical models of drug bioavailability 6.1.2 Outlying observations in pharmacokinetic models 6.2 Oxygen saturation in respiratory physiology 6.2.1 Problem background 6.2.2 The data 6.2.3 Modelling justification 6.2.4 Problem solution 6.3 Production models in Economics 6.3.1 The two-step approach 6.3.1.1 Two-step nonlinear least squares 6.3.1.2 Two-step adaptive Lp-norm estimation 6.3.2 The global least squares approach Concluding remark Bibliography: Chapter 6 AUTHOR INDEX SUBJECT INDEX