Robust statistics: theory and methods
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Chichester
Wiley
2006
|
| Edition: | Reprinted with corr. |
| Series: | Wiley series in probability and statistics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XX, 403 S. graph. Darst. |
| ISBN: | 0470010924 9780470010921 |
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| 100 | 1 | |a Maronna, Ricardo A. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Robust statistics |b theory and methods |c Ricardo A. Maronna ; R. Douglas Martin ; Víctor J. Yohai |
| 250 | |a Reprinted with corr. | ||
| 264 | 1 | |a Chichester |b Wiley |c 2006 | |
| 300 | |a XX, 403 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Wiley series in probability and statistics | |
| 650 | 7 | |a Statistiek |2 gtt | |
| 650 | 4 | |a Statistiques robustes | |
| 650 | 4 | |a Statistik | |
| 650 | 4 | |a Robust statistics | |
| 650 | 0 | 7 | |a Robuste Statistik |0 (DE-588)4451047-0 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Robuste Statistik |0 (DE-588)4451047-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Martin, R. Douglas |e Verfasser |4 aut | |
| 700 | 1 | |a Yohai, Victor J. |e Verfasser |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016306115&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-016306115 | ||
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| adam_text | Contents
1
Introduction
1.1
Classical and robust approaches to statistics
1.2
Mean and standard deviation
1.3
The three-sigma edit rule
1.4
Linear regression
1.4.1
Straight-line regression
1.4.2
Multiple linear regression
1.5
Correlation coefficients
1.6
Other parametric models
1.7
Problems
Preface
xv
1
1
2
5
7
7
9
Π
13
15
2
Location and Scale
17
2.1
The location model
17
2.2
M-estimates of location
22
2.2.1
Generalizing maximum likelihood
22
2.2.2
The distribution of M-estimates
25
2.2.3
An intuitive view of M-estimates
27
2.2.4
Redescending M-estimates
29
2.3
Trimmed means
31
2.4
Dispersion estimates
32
2.5
M-estimates of scale
34
2.6
M-estimates of location with unknown dispersion
36
2.6.1
Previous estimation of dispersion
37
2.6.2
Simultaneous M-estimates of location and dispersion
37
2.7
Numerical computation of M-estimates
39
2.7.1
Location with previously computed dispersion estimation
39
2.7.2
Scale estimates
40
2.7.3
Simultaneous estimation of location and dispersion
41
viii CONTENTS
2.8
Robust
confidence
intervals
and tests
41
2.8.1
Confidence intervals
41
2.8.2
Tests
43
2.9
Appendix: proofs and complements
44
2.9.1
Mixtures
44
2.9.2
Asymptotic normality of M-estimates
45
2.9.3
Slutsky s lemma
46
2.9.4
Quantiles
46
2.9.5
Alternative algorithms for M-estimates
46
2.10
Problems
48
3
Measuring Robustness
51
3.1
The influence function
55
3.1.1
*The convergence of the SC to the IF
57
3.2
The breakdown point
58
3.2.1
Location M-estimates
58
3.2.2
Scale and dispersion estimates
59
3.2.3
Location with previously computed dispersion estimate
60
3.2.4
Simultaneous estimation
60
3.2.5
Finite-sample breakdown point
61
3.3
Maximum asymptotic bias
62
3.4
Balancing robustness and efficiency
64
3.5
* Optimal robustness
65
3.5.1
Bias and variance optimality of location estimates
66
3.5.2
Bias optimality of scale and dispersion estimates
66
3.5.3
The infinitesimal approach
67
3.5.4
The Hampel approach
68
3.5.5
Balancing bias and variance: the general problem
70
3.6
Multidimensional parameters
70
3.7
*Estimates as functionals
71
3.8
Appendix: proofs of results
75
3.8.1
IF of general M-estimates
75
3.8.2
Maximum BP of location estimates
76
3.8.3
BP of location M-estimates
76
3.8.4
Maximum bias of location M-estimates
78
3.8.5
The minimax bias property of the median
79
3.8.6
Minimizing the
GES
80
3.8.7
Hampel optimality
82
3.9
Problems
84
4
Linear Regression
1 87
4.1
Introduction
87
4.2
Review of the LS method
91
4.3
Classical methods for outlier detection
94
CONTENTS ix
4.4 Regression M-estimates 98
4.4.1 M-estimates
with known scale
99
4.4.2 M-estimates
with preliminary scale
100
4.4.3
Simultaneous estimation of regression and scale
103
4.5
Numerical computation of monotone M-estimates
103
4.5.1
The LI estimate
103
4.5.2
M-estimates with smooth i/r-function
104
4.6
Breakdown point of monotone regression estimates
105
4.7
Robust tests for linear hypothesis
107
4.7.1
Review of the classical theory
107
4.7.2
Robust tests using M-estimates
108
4.8
*Regression quantiles
110
4.9
Appendix: proofs and complements
1
Ю
4.9.1
Why equivariance?
110
4.9.2
Consistency of estimated slopes under asymmetric errors 111
4.9.3
Maximum FBP of equivariant estimates
112
4.9.4
The FBP of monotone M-estimates
113
4.10
Problems
114
5
Linear Regression
2 115
5.1
Introduction
115
5.2
The linear model with random predictors
118
5.3
M-estimates with a bounded /j-function
119
5.4
Properties of M-estimates with a bounded p-fimction
120
5.4.1
Breakdown point
122
5.4.2
Influence function
123
5.4.3
Asymptotic normality
123
5.5
MM-estimates
124
5.6
Estimates based on a robust residual scale
126
5.6.1
S-estimates
129
5.6.2
L-estimates of scale and the
LTS
estimate
131
5.6.3
Improving efficiency with one-step reweighting
132
5.6.4
A fully efficient one-step procedure
133
5.7
Numerical computation of estimates based on robust scales
134
5.7.1
Finding local minima
136
5.7.2
The
subsampling
algorithm
136
5.7.3
A strategy for fast iterative estimates
138
5.8
Robust confidence intervals and tests for M-estimates
139
5.8.1
Bootstrap robust confidence intervals and tests
141
5.9
Balancing robustness and efficiency
141
5.9.1
Optimal redescending M-estimates
144
5.10
The exact fit property
146
5.11
Generalized M-estimates
147
5.12
Selection of variables
150
CONTENTS
5.13
Heteroskedastic errors
153
5.13.1
Improving the efficiency of M-estimates
153
5.13.2
Estimating the asymptotic covariance matrix under
heteroskedastic errors
154
5.14
*Other estimates
156
5.14.1
т
-estimates
156
5.14.2
Projection estimates
157
5.14.3
Constrained M-estimates
158
5.14.4
Maximum depth estimates
158
5.15
Models with numeric and categorical predictors
159
5.16 *
Appendix: proofs and complements
162
5.16.1
The BP of monotone M-estimates with random X
162
5.16.2
Heavy-tailed
χ
162
5.16.3
Proof of the exact fit property
163
5.16.4
The BP of S-estimates
163
5.16.5
Asymptotic bias of M-estimates
166
5.16.6
Hampel optimality for GM-estimates
167
5.16.7
Justification of RFPE*
168
5.16.8
A robust multiple correlation coefficient
170
5.17
Problems
171
Multivariate Analysis
175
6.1
Introduction
175
6.2
Breakdown and efficiency of multivariate estimates
180
6.2.1
Breakdown point
180
6.2.2
The multivariate exact fit property
181
6.2.3
Efficiency
181
6.3
M-estimates
182
6.3.1
Collinearity
184
6.3.2
Size and shape
185
6.3.3
Breakdown point
186
6.4
Estimates based on a robust scale
187
6.4.1
The minimum volume ellipsoid estimate
187
6.4.2
S-estimates
188
6.4.3
The minimum covariance determinant estimate
189
6.4.4
S-estimates for high dimension
190
6.4.5
One-step reweighting
193
6.5
The Stahel-Donoho estimate
193
6.6
Asymptotic bias
195
6.7
Numerical computation of multivariate estimates
197
6.7.1
Monotone M-estimates
197
6.7.2
Local solutions for S-estimates
197
6.7.3
Subsampling
for estimates based on a robust scale
198
6.7.4
TheMVE
199
6.7.5
Computation of S-estimates
199
CONTENTS xi
6.7.6
The MCD
200
6.7.7 The Stahel-Donoho
estimate
200
6.8
Comparing estimates
200
6.9
Faster robust dispersion matrix estimates
204
6.9.1
Using pairwise robust covariances
204
6.9.2
Using kurtosis
208
6.10
Robust principal components
209
6.10.1
Robust PCA based on a robust scale
211
6.10.2
Spherical principal components
212
6.11
*Other estimates of location and dispersion
214
6.11.1
Projection estimates
214
6.11.2
Constrained M-estimates
215
6.11.3
Multivariate MM- and
τ
-estimates
216
6.11.4
Multivariate depth
2
1
6
6.12
Appendix: proofs and complements
216
6.12.1
Why
affine
equivariance?
216
6.12.2
Consistency of equivariant estimates
217
6.12.3
The estimating equations of the MLE
217
6.12.4
Asymptotic BP of monotone M-estimates
218
6.12.5
The estimating equations for S-estimates
220
6.12.6
Behavior of S-estimates for high
ρ
221
6.12.7
Calculating the asymptotic covariance matrix of
location M-estimates
222
6.12.8
The exact fit property
224
6.12.9
Elliptical distributions
224
6.12.10
Consistency of Gnanadesikan-Kettenring correlations
225
6.12.11
Spherical principal components
226
6.13
Problems
227
7
Generalized Linear Models
229
7.1
Logistic regression
229
7.2
Robust estimates for the logistic model
233
7.2.1
Weighted MLEs
233
7.2.2
Redescending M-estimates
234
7.3
Generalized linear models
239
7.3.1
Conditionally unbiased bounded influence estimates
242
7.3.2
Other estimates for GLMs
243
7.4
Problems
244
8
Time Series 247
8.1
Time series outliers and their impact
247
8.1.1
Simple examples of outliers influence
250
8.1.2
Probability models for time series outliers
252
8.1.3
Bias impact of AOs
256
xii
CONTENTS
8.2
Classical estimates for
AR
models
257
8.2.1
The Durbin-Levinson algorithm
260
8.2.2
Asymptotic distribution of classical estimates
262
8.3
Classical estimates for
ARMA
models
264
8.4
M-estimates of
ARMA
models
266
8.4.1
M-estimates and their asymptotic distribution
266
8.4.2
The behavior of M-estimates in
AR
processes with AOs
267
8.4.3
The behavior of LS and M-estimates for
ARMA
processes with infinite innovations variance
268
8.5
Generalized M-estimates
270
8.6
Robust
AR
estimation using robust filters
271
8.6.1
Naive minimum robust scale
AR
estimates
272
8.6.2
The robust filter algorithm
272
8.6.3
Minimum robust scale estimates based on robust filtering
275
8.6.4
A robust Durbin-Levinson algorithm
275
8.6.5
Choice of scale for the robust Durbin-Levinson procedure
276
8.6.6
Robust identification of
AR
order
277
8.7
Robust model identification
278
8.7.1
Robust autocorrelation estimates
278
8.7.2
Robust partial autocorrelation estimates
284
8.8
Robust
ARMA
model estimation using robust filters
287
8.8.1
τ
-estimates of
ARMA
models
287
8.8.2
Robust filters for
ARMA
models
288
8.8.3
Robustly filtered r-estimates
290
8.9
ARIMA and SARIMA models
291
8.10
Detecting time series outliers and level shifts
294
8.10.1
Classical detection of time series outliers and level shifts
295
8.10.2
Robust detection of outliers and level shifts for
ARIMA models
297
8.10.3
REGARIMA models: estimation and outlier detection
300
8.11
Robustness measures for time series
301
8.11.1
Influence function
301
8.11.2
Maximum bias
303
8.11.3
Breakdown point
304
8.11.4
Maximum bias curves for the AR(
1 )
model
305
8.12
Other approaches for
ARMA
models
306
8.12.1
Estimates based on robust autocovariances
306
8.12.2
Estimates based on memory-m prediction residuals
308
8.13
High-efficiency robust location estimates
308
8.14
Robust spectral density estimation
309
8.14.1
Definition of the spectral density
309
8.14.2 AR
spectral density
310
8.14.3
Classic spectral density estimation methods
311
8.14.4
Prewhitening
312
CONTENTS
xiii
8.14.5
Influence
of outliers on spectral density estimates
312
8.14.6
Robust spectral density estimation
314
8.14.7
Robust time-average spectral density estimate
316
8.15
Appendix A: heuristic derivation of the asymptotic distribution
of M-estimates for
ARMA
models
317
8.16
Appendix
В
:
robust filter covariance recursions
320
8.17
Appendix C:
ARMA
model state-space representation
322
8.18
Problems
323
9
Numerical Algorithms
325
9.1
Regression M-estimates
325
9.2
Regression S-estimates
328
9.3
The LTS-estimate
328
9.4
Scale M-estimates
328
9.4.1
Convergence of the fixed point algorithm
328
9.4.2
Algorithms for the
nonconcave case
330
9.5
Multivariate M-estimates
330
9.6
Multivariate S-estimates
331
9.6.1
S-estimates with monotone weights
331
9.6.2
The MCD
332
9.6.3
S-estimates with
nonmonotone
weights
333
9.6.4
*Proofof(9.25)
334
10
Asymptotic Theory of M-estimates
335
10.1
Existence and uniqueness of solutions
336
10.2
Consistency
337
10.3
Asymptotic normality
339
10.4
Convergence of the SC to the IF
342
10.5
M-estimates of several parameters
343
10.6
Location M-estimates with preliminary scale
346
10.7
Trimmed means
348
10.8
OptimalityoftheMLE
348
10.9
Regression M-estimates
350
10.9.1
Existence and uniqueness
350
10.9.2
Asymptotic normality: fixed X
351
10.9.3
Asymptotic normality: random X
355
10.10
Nonexistence of moments of the sample median
355
10.11
Problems 356
11
Robust Methods in S-Plus 357
11.1
Location M-estimates: function Mestimate
357
11.2
Robust regression
358
11.2.1
A general function for robust regression: imRob
358
11.2.2
Categorical variables: functions as.factor and contrasts
361
xiv CONTENTS
11.2.3
Testing
linear
assumptions: function rob.linear.test
363
11.2.4
Stepwise variable selection: function step
364
11.3
Robust dispersion matrices
365
11.3.1
A general function for computing robust
location-dispersion estimates: covRob
365
11.3.2
The SR-a estimate: function cov.SRocke
366
11.3.3
The bisquare S-estimate: function cov.Sbic
366
11.4
Principal components
366
11.4.1
Spherical principal components: functionprin.comp.rob
367
11.4.2
Principal components based on a robust dispersion
matrix: function princomp.cov
367
11.5
Generalized linear models
368
11.5.1
M-estimate for logistic models: function BYlogreg
368
11.5.2
Weighted M-estimate: function WBYlogreg
369
11.5.3
A general function for generalized linear models: glmRob
370
11.6
Time series
371
11.6.1
GM-estimates for
AR
models: function ar.gm
371
11.6.2
Ft-estimates and outlier detection for ARIMA and
REGARIMA models: function
arima,
rob
372
11.7
Public-domain software for robust methods
374
12
Description of Data Sets
377
Bibliography
383
Index
397
|
| adam_txt |
Contents
1
Introduction
1.1
Classical and robust approaches to statistics
1.2
Mean and standard deviation
1.3
The "three-sigma edit" rule
1.4
Linear regression
1.4.1
Straight-line regression
1.4.2
Multiple linear regression
1.5
Correlation coefficients
1.6
Other parametric models
1.7
Problems
Preface
xv
1
1
2
5
7
7
9
Π
13
15
2
Location and Scale
17
2.1
The location model
17
2.2
M-estimates of location
22
2.2.1
Generalizing maximum likelihood
22
2.2.2
The distribution of M-estimates
25
2.2.3
An intuitive view of M-estimates
27
2.2.4
Redescending M-estimates
29
2.3
Trimmed means
31
2.4
Dispersion estimates
32
2.5
M-estimates of scale
34
2.6
M-estimates of location with unknown dispersion
36
2.6.1
Previous estimation of dispersion
37
2.6.2
Simultaneous M-estimates of location and dispersion
37
2.7
Numerical computation of M-estimates
39
2.7.1
Location with previously computed dispersion estimation
39
2.7.2
Scale estimates
40
2.7.3
Simultaneous estimation of location and dispersion
41
viii CONTENTS
2.8
Robust
confidence
intervals
and tests
41
2.8.1
Confidence intervals
41
2.8.2
Tests
43
2.9
Appendix: proofs and complements
44
2.9.1
Mixtures
44
2.9.2
Asymptotic normality of M-estimates
45
2.9.3
Slutsky's lemma
46
2.9.4
Quantiles
46
2.9.5
Alternative algorithms for M-estimates
46
2.10
Problems
48
3
Measuring Robustness
51
3.1
The influence function
55
3.1.1
*The convergence of the SC to the IF
57
3.2
The breakdown point
58
3.2.1
Location M-estimates
58
3.2.2
Scale and dispersion estimates
59
3.2.3
Location with previously computed dispersion estimate
60
3.2.4
Simultaneous estimation
60
3.2.5
Finite-sample breakdown point
61
3.3
Maximum asymptotic bias
62
3.4
Balancing robustness and efficiency
64
3.5
*"Optimal" robustness
65
3.5.1
Bias and variance optimality of location estimates
66
3.5.2
Bias optimality of scale and dispersion estimates
66
3.5.3
The infinitesimal approach
67
3.5.4
The Hampel approach
68
3.5.5
Balancing bias and variance: the general problem
70
3.6
Multidimensional parameters
70
3.7
*Estimates as functionals
71
3.8
Appendix: proofs of results
75
3.8.1
IF of general M-estimates
75
3.8.2
Maximum BP of location estimates
76
3.8.3
BP of location M-estimates
76
3.8.4
Maximum bias of location M-estimates
78
3.8.5
The minimax bias property of the median
79
3.8.6
Minimizing the
GES
80
3.8.7
Hampel optimality
82
3.9
Problems
84
4
Linear Regression
1 87
4.1
Introduction
87
4.2
Review of the LS method
91
4.3
Classical methods for outlier detection
94
CONTENTS ix
4.4 Regression M-estimates 98
4.4.1 M-estimates
with known scale
99
4.4.2 M-estimates
with preliminary scale
100
4.4.3
Simultaneous estimation of regression and scale
103
4.5
Numerical computation of monotone M-estimates
103
4.5.1
The LI estimate
103
4.5.2
M-estimates with smooth i/r-function
104
4.6
Breakdown point of monotone regression estimates
105
4.7
Robust tests for linear hypothesis
107
4.7.1
Review of the classical theory
107
4.7.2
Robust tests using M-estimates
108
4.8
*Regression quantiles
110
4.9
Appendix: proofs and complements
1
Ю
4.9.1
Why equivariance?
110
4.9.2
Consistency of estimated slopes under asymmetric errors 111
4.9.3
Maximum FBP of equivariant estimates
112
4.9.4
The FBP of monotone M-estimates
113
4.10
Problems
114
5
Linear Regression
2 115
5.1
Introduction
115
5.2
The linear model with random predictors
118
5.3
M-estimates with a bounded /j-function
119
5.4
Properties of M-estimates with a bounded p-fimction
120
5.4.1
Breakdown point
122
5.4.2
Influence function
123
5.4.3
Asymptotic normality
123
5.5
MM-estimates
124
5.6
Estimates based on a robust residual scale
126
5.6.1
S-estimates
129
5.6.2
L-estimates of scale and the
LTS
estimate
131
5.6.3
Improving efficiency with one-step reweighting
132
5.6.4
A fully efficient one-step procedure
133
5.7
Numerical computation of estimates based on robust scales
134
5.7.1
Finding local minima
136
5.7.2
The
subsampling
algorithm
136
5.7.3
A strategy for fast iterative estimates
138
5.8
Robust confidence intervals and tests for M-estimates
139
5.8.1
Bootstrap robust confidence intervals and tests
141
5.9
Balancing robustness and efficiency
141
5.9.1
"Optimal" redescending M-estimates
144
5.10
The exact fit property
146
5.11
Generalized M-estimates
147
5.12
Selection of variables
150
CONTENTS
5.13
Heteroskedastic errors
153
5.13.1
Improving the efficiency of M-estimates
153
5.13.2
Estimating the asymptotic covariance matrix under
heteroskedastic errors
154
5.14
*Other estimates
156
5.14.1
т
-estimates
156
5.14.2
Projection estimates
157
5.14.3
Constrained M-estimates
158
5.14.4
Maximum depth estimates
158
5.15
Models with numeric and categorical predictors
159
5.16 *
Appendix: proofs and complements
162
5.16.1
The BP of monotone M-estimates with random X
162
5.16.2
Heavy-tailed
χ
162
5.16.3
Proof of the exact fit property
163
5.16.4
The BP of S-estimates
163
5.16.5
Asymptotic bias of M-estimates
166
5.16.6
Hampel optimality for GM-estimates
167
5.16.7
Justification of RFPE*
168
5.16.8
A robust multiple correlation coefficient
170
5.17
Problems
171
Multivariate Analysis
175
6.1
Introduction
175
6.2
Breakdown and efficiency of multivariate estimates
180
6.2.1
Breakdown point
180
6.2.2
The multivariate exact fit property
181
6.2.3
Efficiency
181
6.3
M-estimates
182
6.3.1
Collinearity
184
6.3.2
Size and shape
185
6.3.3
Breakdown point
186
6.4
Estimates based on a robust scale
187
6.4.1
The minimum volume ellipsoid estimate
187
6.4.2
S-estimates
188
6.4.3
The minimum covariance determinant estimate
189
6.4.4
S-estimates for high dimension
190
6.4.5
One-step reweighting
193
6.5
The Stahel-Donoho estimate
193
6.6
Asymptotic bias
195
6.7
Numerical computation of multivariate estimates
197
6.7.1
Monotone M-estimates
197
6.7.2
Local solutions for S-estimates
197
6.7.3
Subsampling
for estimates based on a robust scale
198
6.7.4
TheMVE
199
6.7.5
Computation of S-estimates
199
CONTENTS xi
6.7.6
The MCD
200
6.7.7 The Stahel-Donoho
estimate
200
6.8
Comparing estimates
200
6.9
Faster robust dispersion matrix estimates
204
6.9.1
Using pairwise robust covariances
204
6.9.2
Using kurtosis
208
6.10
Robust principal components
209
6.10.1
Robust PCA based on a robust scale
211
6.10.2
Spherical principal components
212
6.11
*Other estimates of location and dispersion
214
6.11.1
Projection estimates
214
6.11.2
Constrained M-estimates
215
6.11.3
Multivariate MM- and
τ
-estimates
216
6.11.4
Multivariate depth
2
1
6
6.12
Appendix: proofs and complements
216
6.12.1
Why
affine
equivariance?
216
6.12.2
Consistency of equivariant estimates
217
6.12.3
The estimating equations of the MLE
217
6.12.4
Asymptotic BP of monotone M-estimates
218
6.12.5
The estimating equations for S-estimates
220
6.12.6
Behavior of S-estimates for high
ρ
221
6.12.7
Calculating the asymptotic covariance matrix of
location M-estimates
222
6.12.8
The exact fit property
224
6.12.9
Elliptical distributions
224
6.12.10
Consistency of Gnanadesikan-Kettenring correlations
225
6.12.11
Spherical principal components
226
6.13
Problems
227
7
Generalized Linear Models
229
7.1
Logistic regression
229
7.2
Robust estimates for the logistic model
233
7.2.1
Weighted MLEs
233
7.2.2
Redescending M-estimates
234
7.3
Generalized linear models
239
7.3.1
Conditionally unbiased bounded influence estimates
242
7.3.2
Other estimates for GLMs
243
7.4
Problems
244
8
Time Series 247
8.1
Time series outliers and their impact
247
8.1.1
Simple examples of outliers' influence
250
8.1.2
Probability models for time series outliers
252
8.1.3
Bias impact of AOs
256
xii
CONTENTS
8.2
Classical estimates for
AR
models
257
8.2.1
The Durbin-Levinson algorithm
260
8.2.2
Asymptotic distribution of classical estimates
262
8.3
Classical estimates for
ARMA
models
264
8.4
M-estimates of
ARMA
models
266
8.4.1
M-estimates and their asymptotic distribution
266
8.4.2
The behavior of M-estimates in
AR
processes with AOs
267
8.4.3
The behavior of LS and M-estimates for
ARMA
processes with infinite innovations variance
268
8.5
Generalized M-estimates
270
8.6
Robust
AR
estimation using robust filters
271
8.6.1
Naive minimum robust scale
AR
estimates
272
8.6.2
The robust filter algorithm
272
8.6.3
Minimum robust scale estimates based on robust filtering
275
8.6.4
A robust Durbin-Levinson algorithm
275
8.6.5
Choice of scale for the robust Durbin-Levinson procedure
276
8.6.6
Robust identification of
AR
order
277
8.7
Robust model identification
278
8.7.1
Robust autocorrelation estimates
278
8.7.2
Robust partial autocorrelation estimates
284
8.8
Robust
ARMA
model estimation using robust filters
287
8.8.1
τ
-estimates of
ARMA
models
287
8.8.2
Robust filters for
ARMA
models
288
8.8.3
Robustly filtered r-estimates
290
8.9
ARIMA and SARIMA models
291
8.10
Detecting time series outliers and level shifts
294
8.10.1
Classical detection of time series outliers and level shifts
295
8.10.2
Robust detection of outliers and level shifts for
ARIMA models
297
8.10.3
REGARIMA models: estimation and outlier detection
300
8.11
Robustness measures for time series
301
8.11.1
Influence function
301
8.11.2
Maximum bias
303
8.11.3
Breakdown point
304
8.11.4
Maximum bias curves for the AR(
1 )
model
305
8.12
Other approaches for
ARMA
models
306
8.12.1
Estimates based on robust autocovariances
306
8.12.2
Estimates based on memory-m prediction residuals
308
8.13
High-efficiency robust location estimates
308
8.14
Robust spectral density estimation
309
8.14.1
Definition of the spectral density
309
8.14.2 AR
spectral density
310
8.14.3
Classic spectral density estimation methods
311
8.14.4
Prewhitening
312
CONTENTS
xiii
8.14.5
Influence
of outliers on spectral density estimates
312
8.14.6
Robust spectral density estimation
314
8.14.7
Robust time-average spectral density estimate
316
8.15
Appendix A: heuristic derivation of the asymptotic distribution
of M-estimates for
ARMA
models
317
8.16
Appendix
В
:
robust filter covariance recursions
320
8.17
Appendix C:
ARMA
model state-space representation
322
8.18
Problems
323
9
Numerical Algorithms
325
9.1
Regression M-estimates
325
9.2
Regression S-estimates
328
9.3
The LTS-estimate
328
9.4
Scale M-estimates
328
9.4.1
Convergence of the fixed point algorithm
328
9.4.2
Algorithms for the
nonconcave case
330
9.5
Multivariate M-estimates
330
9.6
Multivariate S-estimates
331
9.6.1
S-estimates with monotone weights
331
9.6.2
The MCD
332
9.6.3
S-estimates with
nonmonotone
weights
333
9.6.4
*Proofof(9.25)
334
10
Asymptotic Theory of M-estimates
335
10.1
Existence and uniqueness of solutions
336
10.2
Consistency
337
10.3
Asymptotic normality
339
10.4
Convergence of the SC to the IF
342
10.5
M-estimates of several parameters
343
10.6
Location M-estimates with preliminary scale
346
10.7
Trimmed means
348
10.8
OptimalityoftheMLE
348
10.9
Regression M-estimates
350
10.9.1
Existence and uniqueness
350
10.9.2
Asymptotic normality: fixed X
351
10.9.3
Asymptotic normality: random X
355
10.10
Nonexistence of moments of the sample median
355
10.11
Problems 356
11
Robust Methods in S-Plus 357
11.1
Location M-estimates: function Mestimate
357
11.2
Robust regression
358
11.2.1
A general function for robust regression: imRob
358
11.2.2
Categorical variables: functions as.factor and contrasts
361
xiv CONTENTS
11.2.3
Testing
linear
assumptions: function rob.linear.test
363
11.2.4
Stepwise variable selection: function step
364
11.3
Robust dispersion matrices
365
11.3.1
A general function for computing robust
location-dispersion estimates: covRob
365
11.3.2
The SR-a estimate: function cov.SRocke
366
11.3.3
The bisquare S-estimate: function cov.Sbic
366
11.4
Principal components
366
11.4.1
Spherical principal components: functionprin.comp.rob
367
11.4.2
Principal components based on a robust dispersion
matrix: function princomp.cov
367
11.5
Generalized linear models
368
11.5.1
M-estimate for logistic models: function BYlogreg
368
11.5.2
Weighted M-estimate: function WBYlogreg
369
11.5.3
A general function for generalized linear models: glmRob
370
11.6
Time series
371
11.6.1
GM-estimates for
AR
models: function ar.gm
371
11.6.2
Ft-estimates and outlier detection for ARIMA and
REGARIMA models: function
arima,
rob
372
11.7
Public-domain software for robust methods
374
12
Description of Data Sets
377
Bibliography
383
Index
397 |
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| author | Maronna, Ricardo A. Martin, R. Douglas Yohai, Victor J. |
| author_facet | Maronna, Ricardo A. Martin, R. Douglas Yohai, Victor J. |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | Reprinted with corr. |
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| id | DE-604.BV023103419 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:45:37Z |
| indexdate | 2024-07-09T21:11:05Z |
| institution | BVB |
| isbn | 0470010924 9780470010921 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016306115 |
| oclc_num | 69672553 |
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| owner | DE-703 DE-706 DE-11 |
| owner_facet | DE-703 DE-706 DE-11 |
| physical | XX, 403 S. graph. Darst. |
| publishDate | 2006 |
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| publisher | Wiley |
| record_format | marc |
| series2 | Wiley series in probability and statistics |
| spelling | Maronna, Ricardo A. Verfasser aut Robust statistics theory and methods Ricardo A. Maronna ; R. Douglas Martin ; Víctor J. Yohai Reprinted with corr. Chichester Wiley 2006 XX, 403 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and statistics Statistiek gtt Statistiques robustes Statistik Robust statistics Robuste Statistik (DE-588)4451047-0 gnd rswk-swf Robuste Statistik (DE-588)4451047-0 s DE-604 Martin, R. Douglas Verfasser aut Yohai, Victor J. Verfasser aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016306115&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Maronna, Ricardo A. Martin, R. Douglas Yohai, Victor J. Robust statistics theory and methods Statistiek gtt Statistiques robustes Statistik Robust statistics Robuste Statistik (DE-588)4451047-0 gnd |
| subject_GND | (DE-588)4451047-0 |
| title | Robust statistics theory and methods |
| title_auth | Robust statistics theory and methods |
| title_exact_search | Robust statistics theory and methods |
| title_exact_search_txtP | Robust statistics theory and methods |
| title_full | Robust statistics theory and methods Ricardo A. Maronna ; R. Douglas Martin ; Víctor J. Yohai |
| title_fullStr | Robust statistics theory and methods Ricardo A. Maronna ; R. Douglas Martin ; Víctor J. Yohai |
| title_full_unstemmed | Robust statistics theory and methods Ricardo A. Maronna ; R. Douglas Martin ; Víctor J. Yohai |
| title_short | Robust statistics |
| title_sort | robust statistics theory and methods |
| title_sub | theory and methods |
| topic | Statistiek gtt Statistiques robustes Statistik Robust statistics Robuste Statistik (DE-588)4451047-0 gnd |
| topic_facet | Statistiek Statistiques robustes Statistik Robust statistics Robuste Statistik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016306115&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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