Bootstrap一致性结果中条件与联合无条件weak convergence的注记
A note on conditional versus joint unconditional weak convergence in bootstrap consistency results
摘要：
Bootstrap或Rescant格式的相容性经典地通过条件律的弱收敛来验证。然而，当在有界函数空间中处理随机过程及其在霍夫曼-约根森意义下的弱收敛时，会出现一个障碍：由于可能的不可测性，定律和条件定律都没有很好的定义。从弱收敛的有界Lipschitz度量的基础上的一个等价公式出发，一个经典的规避方法是用（不可测的）bootstrap过程的\emph{conditional law}和极限过程的律之间的距离来表示bootstrap一致性。本文的主要贡献是给出了有界函数空间中自举一致性的一个等价形式，它更直观、更易于处理。从本质上讲，等价的配方包括（无条件）弱收敛的原始过程与两个自助复制。作为一个副产品，我们提供了两个等价的配方的自举一致性的统计量采取的值在可分度量空间：第一个方面的（无条件）弱收敛的统计量连同其自举复制，第二个方面的依概率收敛的经验分布函数的自举复制。最后，简要回顾了基于Bootstrap的置信区间和检验的渐进有效性，特别强调了在实践中不可避免的条件分位数的蒙特卡罗逼近。
英文摘要：
The consistency of a bootstrap or resampling scheme is classically validated by weak convergence of conditional laws. However, when working with stochastic processes in the space of bounded functions and their weak convergence in the Hoffmann-Jørgensen sense, an obstacle occurs: due to possible non-measurability, neither laws nor conditional laws are well-defined. Starting from an equivalent formulation of weak convergence based on the bounded Lipschitz metric, a classical circumvent is to formulate bootstrap consistency in terms of the latter distance between what might be called a \emph{conditional law} of the (non-measurable) bootstrap process and the law of the limiting process. The main contribution of this note is to provide an equivalent formulation of bootstrap consistency in the space of bounded functions which is more intuitive and easy to work with. Essentially, the equivalent formulation consists of (unconditional) weak convergence of the original process jointly with two bootstrap replicates. As a by-product, we provide two equivalent formulations of bootstrap consistency for statistics taking values in separable metric spaces: the first in terms of (unconditional) weak convergence of the statistic jointly with its bootstrap replicates, the second in terms of convergence in probability of the empirical distribution function of the bootstrap replicates. Finally, the asymptotic validity of bootstrap-based confidence intervals and tests is briefly revisited, with particular emphasis on the, in practice unavoidable, Monte Carlo approximation of conditional quantiles.
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