Nonlinear Lp-Norm Estimation

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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
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