In these notes some results are presented for the asymptotic behavior of the bootstrap procedure. Bootstrap is a procedure for estimating (approximating) the distribution of a statistic. It is based on resampling and simulations. It was been introduced in Efron (1979) and in the last decade it has been discussed for a wide variety of statistical problems. Introductory are the articles Efron and Gong (1983) and Efron and Tibshirani (1986) and the book Helmers (1991b). Many applications of bootstrap are discussed in Efron (1982). Survey articles are Beran (1984b), Hinkley (1988), and Diciccio and Romano (1988a). For many classical decision problems (testing and estimation problems, prediction, construction of confidence regions) bootstrap has been compared with classical approximations based on mathematical limit theorems and expansions (for instance normal approximations, empirical Edgeworth expansions) (see for instance Bretagnolle (1983) and Beran (1982, 1984a, 1987, 1988), Abramovitch and Singh (1985), and Hall (1986a, 1988) ). An asymptotic treatment of bootstrap is contained in the book Beran and Ducharme (1991). A detailed analysis of bootstrap based on higher order Edgeworth expansions has been carried out in the book Hall (1992). Recent publications on bootstrap can also be found in the conference volumes LePage and Billard (1992) and Joeckel, Rothe, and Sendler (1992). We will consider the application of bootstrap in three contexts : estimation of smooth functionals, nonparametric curve estimation, and linear models. We do not attempt a complete description of bootstrap in these areas.

Inhalt
0. Introduction.- 1. Bootstrap and Asymptotic Normality.- 1. Introduction.- 2. Bootstrapping linear functionals. The i.i.d. case.- 3. Bootstrapping smooth functionals.- 4. Bootstrap and wild bootstrap in non i.i.d. models.- 5. Some simulations.- 6. Proofs.- Figures.- 2. An Example Where Bootstrap Fails: Comparing Nonparametric Versus Parametric Regression Fits.- 1. A goodness-of-fit test.- 2. How to bootstrap. Bootstrap and wild bootstrap.- 3. Proofs.- 3. A Bootstrap Success Story: Using Nonparametric Density Estimates in K-Sample Problems.- 1. Bootstrap tests.- 2. Bootstrap confidence regions.- 3. Proofs.- 4. A Bootstrap Test on the Number of Modes of a Density.- 1. Introduction.- 2. The number of modes of a kernel density estimator.- 3. Bootstrapping the test statistic.- 4. Proofs.- Figures.- 5. Higher-Order Accuracy of Bootstrap for Smooth Functionals.- 1. Introduction.- 2. Bootstrapping smooth functionals.- 3. Some more simulations. Bootstrapping an M-estimate.- 4. Proof of the theorem.- Figures.- 6. Bootstrapping Linear Models.- 1. Bootstrapping the least squares estimator.- 2. Bootstrapping F-tests.- 3. Proof of Theorem 3.- 7. Bootstrapping Robust Regression.- 1. Introduction.- 2. Bootstrapping M-estimates.- 3. Stochastic expansions of M-estimates.- 4. Proofs.- Figures.- 8. Bootstrap and wild Bootstrap for High-Dimensional Linear Random Design Models.- 1. Introduction.- 2. Consistency of bootstrap for linear contrasts.- 3. Accuracy of the bootstrap.- 4. Bootstrapping F-tests.- 5. Proofs.- Tables.- Figures.- 9. References.