On bootstrap resampling and iteration
On bootstrap resampling and iteration
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DOI：
10.1093/biomet/75.4.661
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发表时间：
1988-12
期刊：
影响因子：
2.7
通讯作者：
P. Hall;Michael A. Martin
中科院分区：
文献类型：
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作者：
P. Hall;Michael A. Martin
总结我们提出了一个统一的方法来引导响应，适用于非常广泛的统计问题。它使人们的注意力集中在一个或多个特征，这是非常重要的，在任何特定的问题，如覆盖误差或长度的置信区间，或偏置点估计。我们的方法很容易，直接导致一个非常普遍的形式的自举迭代，统一和概括目前不同的帐户这个问题。它也为相对复杂的问题提供了简单的解决方案，例如Lehmann（1986）提出的“有条件”短置信区间的建议。我们提出了一个统一的原则，指导操作的自助回归，适用于非常广泛的统计问题，包括偏差减少，收缩，假设检验和置信区间的建设。我们的原则不同于其他方法，因为它直接关注质量或准确性的度量，以方程的形式表达，并寻求其解。一个非常普遍的形式的自助迭代是一个直接的后果，迭代的经验解决方案，以提高该方程的精度。当用于偏差减少时，再分配原则的迭代产生了广义折刀的竞争对手，使偏差减少到任意低的水平。当应用于置信区间时，它产生了Hall（1986）和Beran（1987）的技术。再分配原则很容易导致新的、复杂的问题的解决方案，例如Lehmann（1986）提出的置信区间的经验版本。莱曼认为，一个“理想”的置信区间是一个短的，当它涵盖了真正的参数值，但不一定否则。保留原则提出了一种构造这种区间的简单经验方法。第2节介绍了一般原则，和？3展示了它如何自然地导致自举迭代。在那里，我们表明，在许多问题的实际利益，如偏见减少和覆盖误差减少双边置信区间，每次迭代减少误差的因素n-1，其中n是样本量。在置信区间的情况下，我们的结果使Beran（1987）的结果更加尖锐，Beran（1987）表明，在双侧区间中，覆盖误差减少了n-2倍。我们的n-1规则的主要例外是单侧区间的覆盖误差，其中误差在每次迭代中减少因子n-A。我们的自助迭代方法不仅统一了不同统计问题的迭代哲学，而且统一了同一问题的不同迭代技术。
SUMMARY We propose a single unifying approach to bootstrap resampling, applicable to a very wide range of statistical problems. It enables attention to be focused sharply on one or more characteristics which are of major importance in any particular problem, such as coverage error or length for confidence intervals, or bias for point estimation. Our approach leads easily and directly to a very general form of bootstrap iteration, unifying and generalizing present disparate accounts of this subject. It also provides simple solutions to relatively complex problems, such as a suggestion by Lehmann (1986) for 'conditionally' short confidence intervals. We set out a single unifying principle guiding the operation of bootstrap resampling, applicable to a very wide range of statistical problems including bias reduction, shrinkage, hypothesis testing and confidence interval construction. Our principle differs from other approaches in that it focuses attention directly on a measure of quality or accuracy, expressed in the form of an equation whose solution is sought. A very general form of bootstrap iteration is an immediate consequence of iterating the empirical solution to this equation so as to improve accuracy. When employed for bias reduction, iteration of the resampling principle yields a competitor to the generalized jackknife, enabling bias to be reduced to arbitrarily low levels. When applied to confidence intervals it produces the techniques of Hall (1986) and Beran (1987). The resampling principle leads easily to solutions of new, complex problems, such as empirical versions of confidence intervals proposed by Lehmann (1986). Lehmann argued that an 'ideal' confidence interval is one which is short when it covers the true parameter value but not necessarily otherwise. The resampling principle suggests a simple empirical means of constructing such intervals. Section 2 describes the general principle, and ? 3 shows how it leads naturally to bootstrap iteration. There we show that in many problems of practical interest, such as bias reduction and coverage-error reduction in two-sided confidence intervals, each iteration reduces error by the factor n-1, where n is sample size. In the case of confidence intervals our result sharpens one of Beran (1987), who showed that coverage error is reduced by the factor n-2 in two-sided intervals. The main exception to our n-1 rule is coverage error of one-sided intervals, where error is reduced by the factor n-A at each iteration. Our approach to bootstrap iteration serves to unify not just the philosophy of iteration for different statistical problems, but also different techniques of iteration for the same