Resampling Methods for Dependent Data
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1 S.N. Lahiri Resampling Methods for Dependent Data With 25 Illustrations Springer
2 Contents 1 Scope of Resampling Methods for Dependent Data The Bootstrap Principle Examples Concluding Remarks Notation 13 2 Bootstrap Methods * Introduction IID Bootstrap Inadequacy of IID Bootstrap for Dependent Data Bootstrap Based on IID Innovations Moving Block Bootstrap Nonoverlapping Block Bootstrap Generalized Block Bootstrap Circular Block Bootstrap Stationary Block Bootstrap Subsampling Transformation-Based Bootstrap Sieve Bootstrap 41 3 Properties of Block Bootstrap Methods for the Sample Mean Introduction Consistency of MBB, NBB, CBB: Sample Mean 47
3 XII Contents Consistency of Bootstrap Variance Estimators Consistency of Distribution Function Estimators Consistency of the SB: Sample Mean Consistency of SB Variance Estimators Consistency of SB Distribution Function Estimators 63 4 Extensions and Examples Introduction Smooth Functions of Means M-Estimators Differentiate Functional Bootstrapping the Empirical Process Consistency of the MBB for Differentiable Statistical Functionals Examples 99 5 Comparison of Block Bootstrap Methods Introduction Empirical Comparisons The Theoretical Framework Expansions for the MSEs Theoretical Comparisons Asymptotic Efficiency Comparison at Optimal Block Lengths Concluding Remarks Proofs Proofs of Theorems for the MBB, the NBB, and the CBB Proofs of Theorems for the SB Second-Order Properties Introduction Edgeworth Expansions for the Mean Under Independence Edgeworth Expansions for the Mean Under Dependence Expansions for Functions of Sample Means Expansions Under the Smooth Function Model Under Independence Expansions for Normalized and Studentized Statistics Under Independence 163^ Expansions for Normalized Statistics Under Dependence Expansions for Studentized Statistics Under Dependence Second-Order Properties of Block Bootstrap Methods
4 Contents XIII 7 Empirical Choice of the Block Size Introduction Theoretical Optimal Block Lengths Optimal Block Lengths for Bias and Variance Estimation Optimal Block Lengths for Distribution Function Estimation A Method Based on Subsampling A Nonparametric Plug-in Method Motivation The Bias Estimator The JAB Variance Estimator The Optimal Block Length Estimator Model-Based Bootstrap Introduction Bootstrapping Stationary Autoregressive Processes Bootstrapping Explosive Autoregressive Processes Bootstrapping Unstable Autoregressive Processes Bootstrapping a Stationary ARMA Process Frequency Domain Bootstrap Introduction Bootstrapping Ratio Statistics Spectral Means and Ratio Statistics Frequency Domain Bootstrap for Ratio Statistics Second-Order Correctness of the FDB Bootstrapping Spectral Density Estimators Frequency Domain Bootstrap for Spectral Density Estimation Consistency of the FDB Distribution Function Estimator Bandwidth Selection A Modified FDB Motivation The Autoregressive-Aided FDB A Long-Range Dependence Introduction A Class of Long-Range Dependent Processes Properties of the MBB Method Main Results Proofs Properties of the Subsampling Method Results on the Normalized Sample Mean 252
5 XIV Contents Results on the Studentized Sample Mean Proofs., Numerical Results Bootstrapping Heavy-Tailed Data and Extremes Introduction Heavy-Tailed Distributions Consistency of the MBB Invalidity of the MBB Extremes of Stationary Random Variables Results on Bootstrapping Extremes Bootstrapping Extremes With Estimated Constants Resampling Methods for Spatial Data Introduction Spatial Asymptotic Frameworks Block Bootstrap for Spatial Data on a Regular Grid Description of the Block Bootstrap Method Numerical Examples Consistency of Bootstrap Variance Estimators Results on the Empirical Distribution Function Differentiable Functional Estimation of Spatial Covariance Parameters The Variogram Least Squares Variogram Estimation The RGLS Method Properties of the RGLS Estimators Numerical Examples Bootstrap for Irregularly Spaced Spatial Data A Class of Spatial Stochastic Designs 319" Asymptotic Distribution of M-Estimators A Spatial Block Bootstrap Method Properties of the Spatial Bootstrap Method Resampling Methods for Spatial Prediction Prediction of Integrals Prediction of Point Values - % A 339 B 345 References 349 Author Index 367 Subject Index 371
COPYRIGHTED MATERIAL CONTENTS
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