## 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

Zbl 0706.62017
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