Since Efron's profound paper on the bootstrap, an enormous amount of effort has been spent on the development of bootstrap, jacknife, and other resampling methods. The primary goal of these computer-intensive methods has been to provide statistical tools that work in complex situations without imposing unrealistic or unverifiable assumptions about the data generating mechanism. This book sets out to lay some of the foundations for subsampling methodology and related methods.

Inhalt

I Basic Theory.- 1 Bootstrap Sampling Distributions.- 1.1 Introduction.- 1.1.1 Pivotal Method.- 1.1.2 Asymptotic Pivotal Method.- 1.1.3 Asymptotic Approximation.- 1.1.4 Bootstrap Approximation.- 1.2 Consistency.- 1.3 Case of the Nonparametric Mean.- 1.4 Generalizations to Mean-like Statistics.- 1.5 Bootstrapping the Empirical Process.- 1.6 Differentiability and the Bootstrap.- 1.7 Further Examples.- 1.8 Hypothesis Testing.- 1.9 Conclusions.- 2 Subsampling in the I.I.D. Case.- 2.1 Introduction.- 2.2 The Basic Theorem.- 2.3 Comparison with the Bootstrap.- 2.4 Stochastic Approximation.- 2.5 General Parameters and Other Choices of Root.- 2.5.1 Studentized Roots.- 2.5.2 General Parameter Space.- 2.6 Hypothesis Testing.- 2.7 Data-Dependent Choice of Block Size.- 2.8 Variance Estimation: The Delete-d Jackknife.- 2.9 Conclusions.- 3 Subsampling for Stationary Time Series.- 3.1 Introduction.- 3.2 Univariate Parameter Case.- 3.2.1 Some Motivation: The Simplest Example.- 3.2.2 Theory and Methods for the General Univariate Parameter Case.- 3.2.3 Studentized Roots.- 3.3 Multivariate Parameter Case.- 3.4 Examples.- 3.5 Hypothesis Testing.- 3.6 Data-Dependent Choice of Block Size.- 3.7 Bias Reduction.- 3.8 Variance Estimation.- 3.8.1 General Statistic Case.- 3.8.2 Case of the Sample Mean.- 3.9 Comparison with the Moving Blocks Bootstrap.- 3.10 Conclusions.- 4 Subsampling for Nonstationary Time Series.- 4.1 Introduction.- 4.2 Univariate Parameter Case.- 4.3 Multivariate Parameter Case.- 4.4 Examples.- 4.5 Hypothesis Testing and Data-Dependent Choice of Block Size.- 4.6 Variance Estimation.- 4.7 Conclusions.- 5 Subsampling for Random Fields.- 5.1 Introduction and Definitions.- 5.2 Some Useful Notions of Strong Mixing for Random Fields.- 5.3 Consistency of Subsampling for Random Fields.- 5.3.1 Univariate Parameter Case.- 5.3.2 Multivariate Parameter Case.- 5.4 Variance Estimation and Bias Reduction.- 5.5 Maximum Overlap Subsampling in Continuous Time.- 5.6 Some Illustrative Examples.- 5.7 Conclusions.- 6 Subsampling Marked Point Processes.- 6.1 Introduction.- 6.2 Definitions and Some Different Notions on Mixing.- 6.3 Subsampling Stationary Marked Point Processes.- 6.3.1 Sampling Setup and Assumptions.- 6.3.2 Main Consistency Result.- 6.3.3 Nonstandard Asymptotics.- 6.4 Stochastic Approximation.- 6.5 Variance Estimation via Subsampling.- 6.6 Examples.- 6.7 Conclusions.- 7 Confidence Sets for General Parameters.- 7.1 Introduction.- 7.2 A Basic Theorem for the Empirical Measure.- 7.3 A General Theorem on Subsampling.- 7.4 Subsampling the Empirical Process.- 7.5 Subsampling the Spectral Measure.- 7.6 Conclusions.- II Extensions, Practical Issues, and Applications.- 8 Subsampling with Unknown Convergence Rate.- 8.1 Introduction.- 8.2 Estimation of the Convergence Rate.- 8.2.1 Convergence Rate Estimation: Univariate Parameter Case.- 8.2.2 Convergence Rate Estimation: Multivariate Parameter Case.- 8.3 Subsampling with Estimated Convergence Rate.- 8.4 Conclusions.- 9 Choice of the Block Size.- 9.1 Introduction.- 9.2 Variance Estimation.- 9.2.1 Case of the Sample Mean.- 9.2.2 General Case.- 9.3 Estimation of a Distribution function.- 9.3.1 Calibration Method.- 9.3.2 Minimum Volatility Method.- 9.4 Hypothesis Testing.- 9.4.1 Calibration Method.- 9.4.2 Minimum Volatility Method.- 9.5 Two Simulation Studies.- 9.5.1 Univariate Mean.- 9.5.2 Linear Regression.- 9.6 Conclusions.- 9.7 Tables.- 10 Extrapolation, Interpolation, and Higher-Order Accuracy.- 10.1 Introduction.- 10.2 Background.- 10.3 I.I.D. Data: The Sample Mean.- 10.3.1 Finite Population Correction.- 10.3.2 The Studentized Sample Mean.- 10.3.3 Estimation of a Two-Sided Distribution.- 10.3.4 Extrapolation.- 10.3.5 Robust Interpolation.- 10.4 I.I.D. Data: General Statistics.- 10.4.1 Extrapolation.- 10.4.2 Case of Unknown Convergence Rate to the Asymptotic Approximation.- 10.5 Strong Mixing Data.- 10.5.1 The Studentized Sample Mean.- 10.5.2 Estimation of a Two-Sided Distribution.- 10.5.3 The Unstudentized Sample Mean and the General Extrapolation Result.- 10.5.4 Finite Population Correction in the Mixing Case.- 10.5.5 Bias-Corrected Variance Estimation for Strong Mixing Data.- 10.6 Moderate Deviations in Subsampling Distribution Estimation.- 10.7 Conclusions.- 11 Subsampling the Mean with Heavy Tails.- 11.1 Introduction.- 11.2 Stable Distributions.- 11.3 Extension of Previous Theory.- 11.4 Subsampling Inference for the Mean with Heavy Tails.- 11.4.1 Appealing to a Limiting Stable Law.- 11.4.2 Using Self-Normalizing Sums.- 11.5 Choice of the Block Size.- 11.6 Simulation Study.- 11.7 Conclusions.- 11.8 Tables.- 12 Subsampling the Autoregressive Parameter.- 12.1 Introduction.- 12.2 Extension of Previous Theory.- 12.2.1 The Basic Method.- 12.2.2 Subsampling Studentized Statistics.- 12.3 Subsampling Inference for the Autoregressive Root.- 12.4 Choice of the Block Size.- 12.5 Simulation Study.- 12.6 Conclusions.- 12.7 Tables.- 13 Subsampling Stock Returns.- 13.1 Introduction.- 13.2 Background and Definitions.- 13.2.1 The GMM Approach.- 13.2.2 The VAR Approach.- 13.2.3 A Bootstrap Approach.- 13.3 The Subsampling Approach.- 13.4 Two Simulation Studies.- 13.4.1 Simulating VAR Data.- 13.4.2 Simulating Bootstrap Data.- 13.5 A New Look at Return Regressions.- 13.6 Additional Looks at Return Regressions.- 13.6.1 A Reorganization of Long-Horizon Regressions.- 13.6.2 A Joint Test for Multiple Return Horizons.- 13.7 Conclusions.- 13.8 Tables.- Appendices.- A Some Results on Mixing.- B A General Central Limit Theorem.- References.- Index of Names.- Index of Subjects.