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Exchangeably weighted bootstraps of the general empirical process. (English) Zbl 0792.62038

The bootstrap principle is used to estimate the variance of an estimator from the data. If \(\theta= \theta_ P\) is the parameter of interest, and \(\theta_ n\) is an estimator of \(\theta\) based on the data \(X_ 1, X_ 2,\dots, X_ n\), then the idea is to estimate the unknown distribution by that of \(\widehat{\theta_ n}= \theta_ n (\widehat{X_ 1}, \widehat{X_ 2},\dots, \widehat{X_ n})\), where \(\widehat{X_ i}\) are i.i.d. from the empirical measure \(P_ n\) defined by \(X_ 1, X_ 2,\dots, X_ n\). In many cases the estimator \(\theta_ n\) is a function of the empirical process \(X_ n= n^{1/2}(P_ n -P)\). This process may be indexed by a family of functions, \({\mathcal F}\), and under suitable conditions \(X_ n\) converges weakly to a \(P\)-Brownian bridge indexed by \({\mathcal F}\).
Fix \(\omega\) in the sample space of the \(X_ i\). The bootstrapped empirical function is \[ X_ n= n^{1/2} (\widehat{P_ n}- P_ n^ \omega), \] where \(\widehat{P_ n}\) is computed from an i.i.d. sample from \(P_ n^ \omega\), the empirical function defined by \(X_ 1(\omega), X_ 2(\omega),\dots, X_ n(\omega)\). E. Giné and J. Zinn [ibid. 18, No. 2, 851-869 (1990; Zbl 0706.62017)] established that the bootstrapped empirical process \(\widehat{X_ n}\) also converges to a \(P\)-Brownian bridge under a simple condition on the envelope function on \({\mathcal F}\).
This paper considers bootstrap estimates \(\widehat{P_ n}\) defined by using exchangeable weights to sample \(X_ i(\omega)\) instead of the multinomial weights arising from sampling with replacement. That is, \[ \widehat{P_ n}= n^{-1} \sum_{j=1}^ n W_{nj} \delta_{X_ j^ \omega} \] for \(W=(W_ n j)\) a triangular array of nonnegative random variables with \(\sum_{j=1}^ n W_{nj}=n\).
The authors establish sufficient conditions on \(W\) for the “exchangeably weighted” bootstrap to work, i.e. so that when \({\mathcal F}\) satisfies the conditions of Giné and Zinn, \(\widehat{X_ n}\) converges to a \(P\)- Brownian bridge. The first set of sufficient conditions (in addition to exchangeability and nonnegativity) is that the \(\sup_ n \int_ 0^ \infty (P[| W_{n1}|>t])^{1/2} dt\) be bounded, that the tails of the \(W_{n1}\) be suitably controlled, and that \[ (1/n) \sum_{j=1}^ n (W_{nj}-1)^ 2\to c^ 2>0. \] A second set of conditions are simpler to use when fourth moments of the \(W_{nj}\) exist.
The proofs draw from a number of well-established fields of probability theory and statistics: Hájek’s central limit theorem for rank statistics, several techniques from probability on Banach spaces and empirical processes, reverse martingale convergence, and a variant of an inequality of Hoeffding. There are a number of examples in section 3 which demonstrate the scope of weights that may be considered. These examples include the Bayesian bootstrap, a double bootstrap scheme, several urn models for sampling weights, and deterministic weighting. The last case includes as a special case the delete-\(h\) jackknife procedure for \(h>1\).

MSC:

62G09 Nonparametric statistical resampling methods
60F17 Functional limit theorems; invariance principles
62G30 Order statistics; empirical distribution functions
62E20 Asymptotic distribution theory in statistics
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60G09 Exchangeability for stochastic processes
62G20 Asymptotic properties of nonparametric inference

Citations:

Zbl 0706.62017
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