Part One
Theories
1
1
Robust
Statistics
3
1.1
Robust
Aspects of Data
3
1.2
Robust
Statistics and the Mechanism
for Producing Outliers
4
1.3
Location and
Scale
Parameters
4
1.3.1
Location Parameter
5
1.3.2
Scale Parameters
9
1.3.3
Location and Dispersion
Models
10
1.3.4
Numerical Computation of M-estimates
11
1.4
Redescending M-Estimates
13
1.5
Breakdown
Point
13
1.6
Linear regression
16
1.7
Robust
Approach in Linear Regression
18
1.8
S-Estimator
23
1.9
Least Absolute and Quantile Esimates
25
1.10
Outlier Detection in Linear Regression
26
1.10.1
Studentized and Deletion Studentized Residuals
27
1.10.2
Hadi Potential
27
1.10.3
Elliptic Norm (Cook
Distance)
28
1.10.4
Difference in
Fits
28
1.10.5
Atkinson’s Distance
28
1.10.6
DFBETAS
28
2
Nonlinear Models
31
2.1
Introduction
31
2.2
Basic Concepts
32
2.3
Parameter Estimations
34
2.3.1
Maximum Likelihood Estimators
34
2.3.2
The Ordinary Least Squares Method
36
2.3.3
Generalized Least Squares Estimate
37
2.4
A Nonlinear Model Example
39
3
Robust
Estimators in Nonlinear Regression
41
3.1
Outliers in Nonlinear Regression
41
3.2
Breakdown Point in Nonlinear Regression
42
3.3
Parameter Estimation
44
3.4
Least Absolute and Quantile Estimates
44
3.5
Quantile Regression
44
3.6
Least Median of
Squares
45
3.7
Least Trimmed Squares
47
3.8
Least Trimmed
Differences
48
3.9
S-Estimator
48
3.10
τ
-Estimator
50
3.11
MM-Estimate
50
3.12
Environmental Data Examples
53
3.13
Nonlinear models
54
3.14
Carbon Dioxide Data
60
3.15
Conclusion
61
4
Heteroscedastic Variance
65
4.1
Definitions and Notations
66
4.2
Weighted Regression for
the Nonparametric
Variance Model
67
4.3
Maximum Likelihood Estimates
68
4.4
Variance Modeling and Estimation
70
4.5
Robust
Multistage Estimate
72
4.6
Least Squares Based
Estimate of Variance Parameters
73
4.7
Robust
Least Squares Based Estimate of
the Structural Variance
Parameter
75
4.8
Weighted M-Estimate
76
4.9
Chicken Growth Data Example
77
4.10
Toxicology Data Example
81
4.11
Evaluation and Comparison of Methods
84
5
Autocorrelated Errors
85
5.1
Introduction
85
5.2
Nonlinear Autocorrelated Model
86
5.3
The Classic Two-stage Estimator
87
5.4
Robust
Two-stage Estimator
88
5.5
Economy Data
89
CONTENTS
5
5.6
ARIMA(1,0,1)(0,0,1)7 Autocorrelation Function
93
6
Outlier Detection in
Nonlinear Regression
103
6.1
Introduction
103
6.2
Estimation Methods
104
6.3
Point Influences
105
6.3.1
Tangential Plan Leverage
106
6.3.2
Jacobian Leverage
107
6.3.3
Generalized and Jacobian Leverage
for M-Estimator
108
|
6.4
|
Outlier
|
Detection Measures
|
111
|
|
|
6.4.1
|
Studentized and
Deletion Studentized Residuals
|
112
|
|
|
6.4.2
|
Hadi’s Potential
|
112
|
|
|
6.4.3
|
Elliptic Norm (Cook Distance)
|
113
|
|
|
6.4.4
|
Difference in Fits
|
113
|
|
|
6.4.5
|
Atkinson’s
Distance
|
113
|
|
|
6.4.6
|
DFBETAS
|
114
|
|
|
6.4.7
|
Measures based on
Jacobian and MM-Estimators
|
114
|
|
|
6.4.8
|
Robust Jacobian Leverage and
Local Influences
|
115
|
|
|
6.4.9
|
Overview
|
116
|
6.5
Simulation Study
117
6.6
Numerical Example
118
6.7
Variance Heteroscedasticity
120
6.7.1
Heteroscedastic Variance Studentized Residual
132
6.7.2
Simulation Study, Hetroscedastic Variance
133
6.8
Conclusion
134
Part
Two Computations
137
7
Optimization
139
7.1
Optimization Overview
139
7.2
Iterative Methods
140
7.3
Wolfe Condition
142
7.4
Convergence Criteria
143
7.5
Mixed
Algorithm
143
7.6
Robust
M-Estimator
144
7.7
The Generalized M-Estimator
145
7.8
Some Mathematical Notation
145
7.9
Genetic Algorithm
146
8
nlr
Package
147
8.1
Overview
147
8.2
nl.form
Object
148
8.2.1
selfStart
Initial Values
154
8.3
Model Fit
by
nlr
155