Publishes on Advanced Statistical Methods and Models, Statistical Methods and Inference, Statistical Methods in Clinical Trials. 38 papers and 32.5k citations.
We discuss the following problem: given a random sample $\mathbf{X} = (X_1, X_2, \cdots, X_n)$ from an unknown probability distribution $F$, estimate the sampling distribution of some prespecified random variable $R(\mathbf{X}, F)$, on the basis of the observed data $\mathbf{x}$. (Standard jackknife theory gives an approximate mean and variance in the case $R(\mathbf{X}, F) = \theta(\hat{F}) - \theta(F), \theta$ some parameter of interest.) A general method, called the "bootstrap," is introduced, and shown to work satisfactorily on a variety of estimation problems. The jackknife is shown to be a linear approximation method for the bootstrap. The exposition proceeds by a series of examples: variance of the sample median, error rates in a linear discriminant analysis, ratio estimation, estimating regression parameters, etc.
This is a review of bootstrap methods, concentrating on basic ideas and applications rather than theoretical considerations. It begins with an exposition of the bootstrap estimate of standard error for one-sample situations. Several examples, some involving quite complicated statistical procedures, are given. The bootstrap is then extended to other measures of statistical accuracy such as bias and prediction error, and to complicated data structures such as time series, censored data, and regression models. Several more examples are presented illustrating these ideas. The last third of the paper deals mainly with bootstrap confidence intervals.
Tukey's jackknife estimate of variance for a statistic $S(X_1, X_2, \cdots, X_n)$ which is a symmetric function of i.i.d. random variables $X_i$, is investigated using an ANOVA-like decomposition of $S$. It is shown that the jackknife variance estimate tends always to be biased upwards, a theorem to this effect being proved for the natural jackknife estimate of $\operatorname{Var} S(X_1, X_2, \cdots, X_{n-1})$ based on $X_1, X_2, \cdots, X_n$.
Introduction The Accuracy of a Sample Mean Random Samples and Probabilities The Empirical Distribution Function and the Plug-In Principle Standard Errors and Estimated Standard Errors The Bootstrap Estimate of Standard Error Bootstrap Standard Errors: Some Examples More Complicated Data Structures Regression Models Estimates of Bias The Jackknife Confidence Intervals Based on Bootstrap Tables Confidence Intervals Based on Bootstrap Percentiles Better Bootstrap Confidence Intervals Permutation Tests Hypothesis Testing with the Bootstrap Cross-Validation and Other Estimates of Prediction Error Adaptive Estimation and Calibration Assessing the Error in Bootstrap Estimates A Geometrical Representation for the Bootstrap and Jackknife An Overview of Nonparametric and Parametric Inference Further Topics in Bootstrap Confidence Intervals Efficient Bootstrap Computations Approximate Likelihoods Bootstrap Bioequivalence Discussion and Further Topics Appendix: Software for Bootstrap Computations References