## Resampling Methods for Dependent DataThis 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 |

367 | |

371 | |

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### Common terms and phrases

asymptotic autoregressive process bias block bootstrap estimators block bootstrap methods block size bootstrap version Chapter coefficient conditional distribution conditions of Theorem consider constants convergence converges in distribution covariance data set define the bootstrap definition denote describe distribution function estimator distribution of Tn Edgeworth expansion Efron estimator 6n example following result function H given Hence IID bootstrap iid random variables independent integer Lahiri Lemma level-2 parameter limit distribution long-range dependence M-estimator matrix MBB estimator Note optimal block length P(Tn periodogram polynomials proof of Theorem quantiles random field random variables random vectors ratio statistics real numbers regularity conditions replacing resampled blocks resampling methods Rn(k sample mean sampling distribution sampling region Rn satisfying second-order Section Smooth Function Smooth Function Model spectral density stationary stationary process subregion subsampling method Suppose valid approximation values variogram version of Tn zero

### 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.