Resampling Methods for Dependent Data

by Lahiri, S. N.

• ISBN13: 9780387009285
• ISBN10: 0387009280
• Format: Hardcover
• Copyright: 2003-08-01
• Publisher: Springer Verlag

Summary

This book gives a detailed account of bootstrap methods and their properties for dependent data, covering a wide range of topics such as block bootstrap methods, bootstrap methods in the frequency domain, resampling methods for long range dependent data, and resampling methods for spatial data. The first five chapters of the book treat the theory and applications of block bootstrap methods at the level of a graduate text. The rest of the book is written as a research monograph, with frequent references to the literature, but mostly at a level accessible to graduate students familiar with basic concepts in statistics. Supplemental background material is added in the discussion of such important issues as second order properties of bootstrap methods, bootstrap under long range dependence, and bootstrap for extremes and heavy tailed dependent data. Further, illustrative numerical examples are given all through the book and issues involving application of the methodology are discussed. The book fills a gap in the literature covering research on resampling methods for dependent data that has witnessed vigorous growth over the last two decades but remains scattered in various statistics and econometrics journals. It can be used as a graduate level text for a special topics course on resampling methods for dependent data and also as a research monograph for statisticians and econometricians who want to learn more about the topic and want to apply the methods in their own research. S.N. Lahiri is a professor of Statistics at the Iowa State University, is a Fellow of the Institute of Mathematical Statistics and a Fellow of the American Statistical Association.

Author Biography

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.

Table of Contents

1 Scope of Resampling Methods for Dependent Data	1	(16)
1.1 The Bootstrap Principle	1	(6)
1.2 Examples	7	(5)
1.3 Concluding Remarks	12	(1)
1.4 Notation	13	(4)
2 Bootstrap Methods	17	(28)
2.1 Introduction	17	(1)
2.2 IID Bootstrap	17	(4)
2.3 Inadequacy of TID Bootstrap for Dependent Data	21	(2)
2.4 Bootstrap Based on TID Innovations	23	(7)
2.5 Moving Block Bootstrap
2.6 Nonoverlapping Block Bootstrap	30	(1)
2.7 Generalized Block Bootstrap	31	(6)
2.7.1 Circular Block Bootstrap	33	(1)
2.7.2 Stationary Block Bootstrap	34	(3)
2.8 Subsampling	37	(3)
2.9 Transformation-Based Bootstrap	40	(1)
2.10 Sieve Bootstrap	41	(4)
3 Properties of Block Bootstrap Methods for the Sample Mean	45	(28)
3.1 Introduction	45	(2)
3.2 Consistency of MBB, NBB, CBB: Sample Mean	47	(10)
3.2.1 Consistency of Bootstrap Variance Estimators	48	(6)
3.2.2 Consistency of Distribution Function Estimators	54	(3)
3.3 Consistency of the SB: Sample Mean	57	(16)
3.3.1 Consistency of SB Variance Estimators	57	(6)
3.3.2 Consistency of SB Distribution Function Estimators	63	(10)
4 Extensions and Examples	73	(42)
4.1 Introduction	73	(1)
4.2 Smooth Functions of Means	73	(8)
4.3 M-Estimators	81	(9)
4.4 Differentiable Functionals	90	(9)
4.4.1 Bootstrapping the Empirical Process	92	(2)
4.4.2 Consistency of the MBB for Differentiable Statistical Functionals	94	(5)
4.5 Examples	99	(16)
5 Comparison of Block Bootstrap Methods	115	(30)
5.1 Introduction	115	(1)
5.2 Empirical Comparisons	116	(2)
5.3 The Theoretical Framework	118	(2)
5.4 Expansions for the MSEs	120	(3)
5.5 Theoretical Comparisons	123	(3)
5.5.1 Asymptotic Efficiency	123	(1)
5.5.2 Comparison at Optimal Block Lengths	124	(2)
5.6 Concluding Remarks	126	(1)
5.7 Proofs	127	(18)
5.7.1 Proofs of Theorems 5.1-5.2 for the MBB, the NBB, and the CBB	128	(7)
5.7.2 Proofs of Theorems 5.1-5.2 for the SB	135	(10)
6 Second-Order Properties	145	(30)
6.1 Introduction	145	(2)
6.2 Edgeworth Expansions for the Mean Under Independence	147	(7)
6.3 Edgeworth Expansions for the Mean Under Dependence	154	(6)
6.4 Expansions for Functions of Sample Means	160	(8)
6.4.1 Expansions Under the Smooth Function Model Under Independence	160	(3)
6.4.2 Expansions for Normalized and Studentized Statistics Under Independence	163	(1)
6.4.3 Expansions for Normalized Statistics Under Dependence	164	(2)
6.4.4 Expansions for Studentized Statistics Under Dependence	166	(2)
6.5 Second-Order Properties of Block Bootstrap Methods	168	(7)
7 Empirical Choice of the Block Size	175	(24)
7.1 Introduction	175	(1)
7.2 Theoretical Optimal BlocLengths	175	(7)
7.2.1 Optimal Block Lengths for Bias and Variance Estimation	177	(2)
7.2.2 Optimal Block Lengths for Distribution Function Estimation	179	(3)
7.3 A Method Based on Subsampling	182	(4)
7.4 A Nonparametric Plug-in Method	186	(13)
7.4.1 Motivation	187	(1)
7.4.2 The Bias Estimator	188	(1)
7.4.3 The JAB Variance Estimator	189	(4)
7.4.4 Tb The Optimal Block Length Estimator	193	(6)
8 Model-Based Bootstrap	199	(22)
8.1 Introduction	199	(1)
8.2 Bootstrapping Stationary Autoregressive Processes	200	(5)
8.3 Bootstrapping Explosive Autoregressive Processes	205	(4)
8.4 Bootstrapping Unstable Autoregressive Processes	209	(5)
8.5 Bootstrapping a Stationary ARMA Process	214	(7)
9 Frequency Domain Bootstrap	221	(20)
9.1 Introduction	221	(1)
9.2 Bootstrapping Ratio Statistics	222	(6)
9.2.1 Spectral Means and Ratio Statistics	222	(2)
9.2.2 Frequency Domain Bootstrap for Ratio Statistics	224	(2)
9.2.3 Second-Order Correctness of the FDB	226	(2)
9.3 Bootstrapping Spectral Density Estimators	228	(7)
9.3.1 Frequency Domain Bootstrap for Spectral Density Estimation	229	(2)
9.3.2 Consistency of the FDB Distribution Function Estimator	231	(2)
9.3.3 Bandwidth Selection	233	(2)
9.4 A Modified FDB	235	(6)
9.4.1 Motivation	236	(1)
9.4.2 The Autoregressive-Aided	237	(4)
10 Long-Range Dependence	241	(20)
10.1 Introduction	241	(1)
10.2 A Class of Long-Range Dependent Processes	242	(2)
10.3 Properties of the MBB Method	244	(7)
10.3.1 Main Results	244	(2)
10.3.2 Proofs	246	(5)
10.4 Properties of the Subsampling Method	251	(6)
10.4.1 Results on the Normalized Sample Mean	252	(1)
10.4.2 Results on the Studentized Sample Mean	253	(2)
10.4.3 Proofs	255	(2)
10.5 Numerical Results	257	(4)
11 Bootstrapping Heavy-Tailed Data and Extremes	261	(20)
11.1 Introduction	261	(1)
11.2 Heavy-Tailed Distribution	262	(3)
11.3 Consistency of the MBB	265	(3)
11.4 Invalidity of the MBB	268	(3)
11.5 Extremes of Stationary Random Variables	271	(3)
11.6 Results on Extremes Bootstrapping	274	(3)
11.7 Bootstrapping Extremes With Estimated Constants	277	(4)
12 Resampling Methods for Spatial Data	281	(57)
12.1 Introduction	281	(1)
12.2 Spatial Asymptotic Frameworks	282
12.3 Block Bootstrap for Spatial Data on a Regular Grid	203	(104)
12.3.1 Description of the Block Bootstrap Method	284
12.3.2 Numerical Examples	280	(12)
12.3.3 Consistency of Bootstrap Variance Estimators	292	(9)
12.3.4 Results on the Empirical Distribution Function	301	(3)
12.3.5 Differentiable Functionals	304	(3)
12.4 Estimation of Spatial Covariance Parameters	307	(12)
12.4.1 The Variogram	307	(1)
12.4.2 Least Squares Variogram Estimation	308	(2)
12.4.3 The 8GLS Method	310	(2)
12.4.4 Properties of the RGLS Estimators	312	(3)
12.4.5 Numerical Examples	315	(4)
12.5 Bootstrap for Irregularly Spaced Spatial Data	319	(9)
12.5.1 A Class of Spatial Stochastic Designs	319	(9)
12.5.2 Asymptotic Distribution of M-Estimators	328
12.5.3 A Spatial Block Bootstrap Method	323	(2)
12.5.4 Properties of the Spatial Bootstrap Method	325	(3)
12.6 Resampling Methods for Spatial Prediction	328	(10)
12.6.1 Prediction of Integrals	328	(7)
12.6.2 Prediction of Point Values	335	(3)
A	338	(7)
B	345	(4)
References	349	(18)
Author Index	367	(4)
Subject Index	371