It is proved that the bootstrapped central limit theorem for empirical processes indexed by a class of functions $\mathscr{F}$ and based on a probability measure $P$ holds a.s. if and only if $\mathscr{F} \in \mathrm{CLT}(P)$ and $\int F^2 dP < \infty$, where $F = \sup_{f \in \mathscr{F}}|f|$, and it holds in probability if and only if $\mathscr{F} \in \mathrm{CLT}(P)$. Thus, for a large class of statistics, no local uniformity of the CLT (about $P$) is needed for the bootstrap to work. Consistency of the bootstrap (the bootstrapped law of large numbers) is also characterized. (These results are proved under certain weak measurability assumptions on $\mathscr{F}$.)

/pdf/bootstrapping-general-empirical-measures-2avqho2qrs.pdf

Bootstrapping General Empirical Measures

SUMMARY Approximate confidence sets for a parameter 0 may be obtained by referring a function of 0 and of the sample to an estimated quantile of that function's sampling distribution. We call this function the root of the confidence set. Either asymptotic theory or bootstrap methods can be used to estimate the desired quantile. When the root is not a pivot, in the sense of classical statistics, the actual level of the approximate confidence set may differ substantially from the intended level. Prepivoting is the transformation of a confidence set root by its estimated bootstrap cumulative distribution function. Prepivoting can be iterated. Bootstrap confidence sets generated from a root prepivoted one or more times have smaller error in level than do confidence sets based on the original root. The first prepivoting is nearly equivalent to studentizing, when that operation is appropriate. Further iterations of prepivoting make higher order corrections automatically.

Prepivoting to reduce level error of confidence sets

On passe en revue plusieurs methodes distinctes basees sur le bootstrap pour construire des intervalles de confiance

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1988.tb01732.x

A Review of Bootstrap Confidence Intervals

The Stahel-Donoho estimators (t, V) of multivariate location and scatter are defined as a weighted mean and a weighted covariance matrix with weights of the form w(r), where w is a weight function and r is a measure of “outlyingness,” obtained by considering all univariate projections of the data. It has a high breakdown point for all dimensions and order √n consistency. The asymptotic bias of V for point mass contamination for suitable weight functions is compared with that of Rousseeuw's minimum volume ellipsoid (MVE) estimator. A simulation shows that for a suitable w, t and V exhibit high efficiency for both normal and Cauchy distributions and are better than their competitors for normal data with point-mass contamination. The performances of the estimators for detecting outliers are compared for both a real and a synthetic data set.

The Behavior of the Stahel-Donoho Robust Multivariate Estimator

Several tests based on the empirical measure have been proposed to test independence of variables, goodness of fit, equality of distributions, rotational invariance, and so forth. These tests have excellent power properties, but critical values are difficult, if not impossible, to obtain. Furthermore, these tests usually assume that the data are real-valued with continuous distributions. Here, critical values are determined by bootstrapping and the resulting tests are shown to have the correct asymptotic level under minimal assumptions. For example, given data Xi = (X i,1, …, Xi,d ), i = 1, …, n, it may be desired to test independence of the d components. The proposed test compares the empirical measure and the product of its marginals by taking a supremum over an appropriate Vapnik-Cervonenkis class of sets. No assumptions are made on the probability distribution of the data or on the space in which it lives; indeed, some components may be discrete, some continuous, and others categorical. Simil...

A Bootstrap Revival of Some Nonparametric Distance Tests

The confidence sets for a $q$-dimensional distribution studied in this paper have several attractive features: affine invariance, correct asymptotic level whatever the actual distribution may be, numerical feasibility, and a local asymptotic minimax optimality property. When dimension $q$ equals one, the confidence sets reduce to the usual Kolmogorov-Smirnov confidence bands, except that critical values are determined by bootstrapping.

/pdf/confidence-sets-for-a-multivariate-distribution-3y2zadpgfp.pdf

R. Beran

Papers

Confidence sets for a multivariate distribution