Resampling Methods for Dependent Data

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Springer Science & Business Media, Aug 7, 2003 - Mathematics - 374 pages
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This is a book on bootstrap and related resampling methods for temporal and spatial data exhibiting various forms of dependence. Like the resam pling methods for independent data, these methods provide tools for sta tistical analysis of dependent data without requiring stringent structural assumptions. This is an important aspect of the resampling methods in the dependent case, as the problem of model misspecification is more preva lent under dependence and traditional statistical methods are often very sensitive to deviations from model assumptions. Following the tremendous success of Efron's (1979) bootstrap to provide answers to many complex problems involving independent data and following Singh's (1981) example on the inadequacy of the method under dependence, there have been several attempts in the literature to extend the bootstrap method to the dependent case. A breakthrough was achieved when resampling of single observations was replaced with block resampling, an idea that was put forward by Hall (1985), Carlstein (1986), Kiinsch (1989), Liu and Singh (1992), and others in various forms and in different inference problems. There has been a vig orous development in the area of res amp ling methods for dependent data since then and it is still an area of active research. This book describes various aspects of the theory and methodology of resampling methods for dependent data developed over the last two decades. There are mainly two target audiences for the book, with the level of exposition of the relevant parts tailored to each audience.
 

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Contents

Scope of Resampling Methods for Dependent Data
1
12 Examples
7
13 Concluding Remarks
12
14 Notation
13
Bootstrap Methods
17
23 Inadequacy of IID Bootstrap for Dependent Data
21
24 Bootstrap Based on IID Innovations
23
25 Moving Block Bootstrap
25
83 Bootstrapping Explosive Autoregressive Processes
205
84 Bootstrapping Unstable Autoregressive Processes
209
85 Bootstrapping a Stationary ARMA Process
214
Frequency Domain Bootstrap
221
92 Bootstrapping Ratio Statistics
222
922 Frequency Domain Bootstrap for Ratio Statistics
224
923 SecondOrder Correctness of the FDB
226
93 Bootstrapping Spectral Density Estimators
228

26 Nonoverlapping Block Bootstrap
30
27 Generalized Block Bootstrap
31
271 Circular Block Bootstrap
33
272 Stationary Block Bootstrap
34
28 Subsampling
37
29 TransformationBased Bootstrap
40
210 Sieve Bootstrap
41
Properties of Block Bootstrap Methods for the Sample Mean
45
Sample Mean
47
321 Consistency of Bootstrap Variance Estimators
48
322 Consistency of Distribution Function Estimators
54
Sample Mean
57
332 Consistency of SB Distribution Function Estimators
63
Extensions and Examples
73
43 MEstimators
81
44 Differentiable Functionals
90
441 Bootstrapping the Empirical Process
92
442 Consistency of the MBB for Differentiable Statistical Functionals
94
45 Examples
99
Comparison of Block Bootstrap Methods
115
52 Empirical Comparisons
116
53 The Theoretical Framework
118
54 Expansions for the MSEs
120
55 Theoretical Comparisons
123
552 Comparison at Optimal Block Lengths
124
56 Concluding Remarks
126
57 Proofs
127
571 Proofs of Theorems 5152 for the MBB the NBB and the CBB
128
572 Proofs of Theorems 5152 for the SB
135
SecondOrder Properties
145
62 Edgeworth Expansions for the Mean Under Independence
147
63 Edgeworth Expansions for the Mean Under Dependence
154
64 Expansions for Functions of Sample Means
160
642 Expansions for Normalized and Studentized Statistics Under Independence
163
643 Expansions for Normalized Statistics Under Dependence
164
644 Expansions for Studentized Statistics Under Dependence
166
65 SecondOrder Properties of Block Bootstrap Methods
168
Empirical Choice of the Block Size
175
721 Optimal Block Lengths for Bias and Variance Estimation
177
722 Optimal Block Lengths for Distribution Function Estimation
179
73 A Method Based on Subsampling
182
74 A Nonparametric Plugin Method
186
741 Motivation
187
742 The Bias Estimator
188
743 The JAB Variance Estimator
189
744 The Optimal Block Length Estimator
193
ModelBased Bootstrap
199
82 Bootstrapping Stationary Autoregressive Processes
200
931 Frequency Domain Bootstrap for Spectral Density Estimation
229
932 Consistency of the FDB Distribution Function Estimator
231
933 Bandwidth Selection
233
94 A Modified FDB
235
941 Motivation
236
942 The Autoregressive Aided FDB
237
LongRange Dependence
241
102 A Class of LongRange Dependent Processes
242
103 Properties of the MBB Method
244
1032 Proofs
246
104 Properties of the Subsampling Method
251
1041 Results on the Normalized Sample Mean
252
1042 Results on the Studentized Sample Mean
253
1043 Proofs
255
105 Numerical Results
257
Bootstrapping HeavyTailed Data and Extremes
261
112 HeavyTailed Distributions
262
113 Consistency of the MBB
265
114 Invalidity of the MBB
268
115 Extremes of Stationary Random Variables
271
116 Results on Bootstrapping Extremes
274
117 Bootstrapping Extremes With Estimated Constants
277
Resampling Methods for Spatial Data
281
122 Spatial Asymptotic Frameworks
282
123 Block Bootstrap for Spatial Data on a Regular Grid
283
1231 Description of the Block Bootstrap Method
284
1232 Numerical Examples
288
1233 Consistency of Bootstrap Variance Estimators
292
1234 Results on the Empirical Distribution Function
301
1235 Differentiable Functionals
304
124 Estimation of Spatial Covariance Parameters
307
1242 Least Squares Variogram Estimation
308
1243 The RGLS Method
310
1244 Properties of the RGLS Estimators
312
1245 Numerical Examples
315
125 Bootstrap for Irregularly Spaced Spatial Data
319
1252 Asymptotic Distribution of MEstimators
320
1253 A Spatial Block Bootstrap Method
323
1254 Properties of the Spatial Bootstrap Method
325
126 Resampling Methods for Spatial Prediction
328
1262 Prediction of Point Values
335
Appendix A
339
Appendix B
345
References
349
Author Index
367
Subject Index
371
Copyright

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Popular passages

Page 360 - A general resampling scheme for triangular arrays of a-mixing random variables with application to the problem of spectral density estimation.
Page 356 - moving block' bootstrap for stationary and nonstationary data", in: R.
Page 360 - Liu, RY and K. Singh, 1992, Moving blocks jackknife and bootstrap capture weak dependence, in: R. LePage, and L. Billard, eds., Exploring the Limits of Bootstrap, (Wiley, New york) 225-248. Nankervis, JC and NE Savin, 1994, The level and power of the bootstrap t-test in the AR(1) model with trend, manuscript, Department of Economics, University of Surrey and University of Iowa.
Page 355 - Department of Agricultural and Resource Economics, North Carolina State University, Raleigh, NC.
Page 363 - Weak convergence of multidimensional empirical processes for strong mixing sequences of stochastic vectors', Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 33, 133-137.
Page 356 - Kreiss, JP and Franke, J. (1992), 'Bootstrapping stationary autoregressive moving-average models', Journal of Time Series Analysis 13, 297-317. Kreiss, JP and Paparoditis, E. (2003), 'Autoregressive aided periodogram bootstrap for time series', Annals of Statistics 31.

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About the author (2003)

S. N. Lahiri is a professor of statistics at the Iowa State University and a fellow of the Institute of Mathematical Statistics and the American Statistical Association.

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