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1995 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

The Jackknife and Bootstrap

verfasst von: Jun Shao, Dongsheng Tu

Verlag: Springer New York

Buchreihe : Springer Series in Statistics

Enthalten in: Professional Book Archive

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Über dieses Buch

The jackknife and bootstrap are the most popular data-resampling meth­ ods used in statistical analysis. The resampling methods replace theoreti­ cal derivations required in applying traditional methods (such as substitu­ tion and linearization) in statistical analysis by repeatedly resampling the original data and making inferences from the resamples. Because of the availability of inexpensive and fast computing, these computer-intensive methods have caught on very rapidly in recent years and are particularly appreciated by applied statisticians. The primary aims of this book are (1) to provide a systematic introduction to the theory of the jackknife, the bootstrap, and other resampling methods developed in the last twenty years; (2) to provide a guide for applied statisticians: practitioners often use (or misuse) the resampling methods in situations where no theoretical confirmation has been made; and (3) to stimulate the use of the jackknife and bootstrap and further devel­ opments of the resampling methods. The theoretical properties of the jackknife and bootstrap methods are studied in this book in an asymptotic framework. Theorems are illustrated by examples. Finite sample properties of the jackknife and bootstrap are mostly investigated by examples and/or empirical simulation studies. In addition to the theory for the jackknife and bootstrap methods in problems with independent and identically distributed (Li.d.) data, we try to cover, as much as we can, the applications of the jackknife and bootstrap in various complicated non-Li.d. data problems.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
The basic objective of statistical analysis is “extracting all the information from the data” (Rao, 1989) to deduce properties of the population that generated the data. Statistical analyses are generally based on statistics, which are functions of data and are selected according to some principle (e.g., the likelihood principle, the substitution principle, sufficiency, and robustness). For example, the sample mean is an estimate of the population center; studentized statistics are used for constructing confidence sets and testing hypotheses in statistical inference
Jun Shao, Dongsheng Tu
Chapter 2. Theory for the Jackknife
Abstract
This chapter presents theory for the jackknife in the case where the data are i.i.d. Many results can be extended in a straightforward manner to more complicated cases, which will be studied in later chapters. We begin this chapter by first focusing on jackknife variance estimators. The basic theoretical consideration in using jackknife variance estimators is their consistency, which is especially crucial when the jackknife variance estimators are used in large sample statistical inference problems such as constructing confidence sets for some unknown parameters. A complete theory for the consistency of the jackknife variance estimators is given in Sections 2.1 and 2.2. We will show that the success of the jackknife variance estimator for a given statistic T n relies on the smoothness of T n , which can be characterized by the differentiability of the function that generates Tn. The jackknife variance estimator may be inconsistent for a statistic that is not very smooth; however, its inconsistency can be rectified by using the delete-d jackknife, an extended version of the jackknife that removes more than one datum at a time (Section 2.3). The delete-d jackknife also provides a jackknife estimator of the sampling distribution of T n , known as the jackknife histogram. Other applications of the jackknife, such as bias estimation and bias reduction, are discussed in Section 2.4. Some empirical results are given as examples.
Jun Shao, Dongsheng Tu
Chapter 3. Theory for the Bootstrap
Abstract
The bootstrap is a very convenient and appealing tool for statistical analysis, however, theoretical and/or empirical confirmation should be made of its suitability for the problem at hand. Also, it is important to know the relative performance of the bootstrap versus other existing methods. A general theory for the bootstrap distribution and variance estimation for a given statistic is presented in this chapter. The more delicate problem of constructing bootstrap confidence sets and hypothesis tests will be treated in the next chapter.
Jun Shao, Dongsheng Tu
Chapter 4. Bootstrap Confidence Sets and Hypothesis Tests
Abstract
In this chapter, we study the applications of the bootstrap in two main components of statistical inference: constructing confidence sets and testing hypotheses. Five different bootstrap confidence sets are introduced in Section 4.1. Asymptotic properties and asymptotic comparisons of these five bootstrap confidence sets and the confidence sets obtained by normal approximation are given in Section 4.2. More sophisticated techniques for bootstrap confidence sets have been developed in recent years, some of which are described in Section 4.3. Fixed sample comparisons of various confidence sets are made in Section 4.4 through empirical simulation studies. Bootstrap hypothesis tests are introduced in Section 4.5.
Jun Shao, Dongsheng Tu
Chapter 5. Computational Methods
Abstract
The jackknife and the bootstrap are computer-intensive methods. Modern computers can meet many computing needs in applying these methods. However, research on the computation of the jackknife and bootstrap estimators is still important, because of the following reasons:
(1)
There are cases (e.g., the delete-d jackknife and the iterative bootstrap) where the computation is cumbersome or even impractical.
 
(2)
More efficient methods for computation are always welcome for saving time and computing expenses.
 
(3)
Questions such as how many bootstrap data sets should be taken in bootstrap Monte Carlo approximations are often asked by practitioners before they apply the bootstrap method.
 
Jun Shao, Dongsheng Tu
Chapter 6. Applications to Sample Surveys
Abstract
A crucial part of sample survey theory is the derivation of a suitable estimator of the variance of a given estimator. The variance estimator can be used in measuring the uncertainty in the estimation, in comparing the efficiencies of sampling designs, and in determining the allocation and stratification under a specific design. It is a common practice to report the estimates in a tabular form along with the variance estimates or estimates of coefficient of variation. Furthermore, consistent variance estimators provide us with large sample confidence sets for the unknown parameters of interest. Confidence sets can also be constructed by directly applying the bootstrap.
Jun Shao, Dongsheng Tu
Chapter 7. Applications to Linear Models
Abstract
In this chapter, we study the application of the jackknife and the bootstrap to linear models. Section 7.1 describes the forms of linear models and estimation of regression parameters in the models. Variance and bias estimation for the estimators of regression parameters are considered in Section 7.2. Other statistical inferences and analyses based on the bootstrap are discussed in Section 7.3. Model selection using cross-validation (jackknife) or the bootstrap is studied in Section 7.4. Most of the technical details and rigorous proofs of the asymptotic results are deferred to Section 7.5.
Jun Shao, Dongsheng Tu
Chapter 8. Applications to Nonlinear, Nonparametric, and Multivariate Models
Abstract
Applications of the jackknife and the bootstrap to various nonlinear parametric (or semiparametric) models, nonparametric curve estimation problems, and multivariate analysis are discussed in this chapter. We mainly focus on the jackknife variance estimation, bootstrap distribution estimation, bootstrap confidence sets, cross-validation (jackknife) and bootstrap model selection, bandwidth (smoothing parameter) selection, and misclas-sification rate estimation. Other applications, such as the jackknife bias estimation and bias reduction, and the bootstrap variance estimation, can be similarly discussed but are omitted.
Jun Shao, Dongsheng Tu
Chapter 9. Applications to Time Series and Other Dependent Data
Abstract
In the last three chapters, we discussed many applications of the jackknife and bootstrap methods in problems with non-i.i.d. data. In most of the problems previously studied, however, the data were either independent or had a cluster structure in which the observations from different clusters were independent and the number of clusters was large so that the jackknife and bootstrap could be applied to clusters instead of the original units that were correlated within each cluster (e.g., the survey data described in Chapter 6 or the longitudinal data described in Chapter 8). In this chapter, we study the application of the jackknife and bootstrap to time series and other dependent data. A time series is a sequence of observations that are indexed by time and are usually correlated. Other dependent data include m-dependent data, Markov chains, α-mixing data, and other stationary stochastic processes that are not indexed by time.
Jun Shao, Dongsheng Tu
Chapter 10. Bayesian Bootstrap and Random Weighting
Abstract
Bayesian bootstrap and random weighting are two variants of the bootstrap. The former is aimed at simulating the posterior distribution of a parameter, given the observations. The latter is still used to estimate the sampling distribution of a random variable but adopts a resampling plan different from the bootstrap: instead of generating resamples from data, the random weighting method assigns a random weight to each observation. Random weighting can be regarded as a smoothing for the bootstrap. These two methods are pooled together and introduced in one chapter because their prime forms are exactly the same and they can be unified into a general framework.
Jun Shao, Dongsheng Tu
Backmatter
Metadaten
Titel
The Jackknife and Bootstrap
verfasst von
Jun Shao
Dongsheng Tu
Copyright-Jahr
1995
Verlag
Springer New York
Electronic ISBN
978-1-4612-0795-5
Print ISBN
978-1-4612-6903-8
DOI
https://doi.org/10.1007/978-1-4612-0795-5