Prepivoting Test Statistics: A Bootstrap View of Asymptotic Refinements
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检验统计量的预枢轴化：渐近改进的 Bootstrap 视角
- 研究问题
- 复合原假设检验中，将统计量通过 Bootstrap 零分布变换，何时能够降低检验水平误差的渐近阶数？
- 方法与思路
- 作者以 Bootstrap 零分布的累积分布函数变换统计量，并比较单次与两次预枢轴化及其蒙特卡洛实现。
- 材料陈述的贡献
- 摘要称，单次变换的改进取决于渐近零分布是否含未知参数；两次变换在正则条件下进一步降低水平误差阶数。
- 阅读关注点
- 应核对正则条件与零分布参数依赖的区别、两次变换的误差阶数，以及嵌套 Bootstrap 的模拟精度和成本。
- 阅读边界
- 仅据版本未注明的 OpenAlex 摘要；正则条件与具体误差阶数未列出，渐近改进不保证所有有限样本均改善。
原文摘要
Approximate tests for a composite null hypothesis about a parameter θ may be obtained by referring a test statistic to an estimated critical value. Either asymptotic theory or bootstrap methods can be used to estimate the desired quantile. The simple asymptotic test ϕA refers the test statistic to a quantile of its asymptotic null distribution after unknown parameters have been estimated. The bootstrap approach used here is based on the concept of prepivoting. Prepivoting is the transformation of a test statistic by the cdf of its bootstrap null distribution. The simple bootstrap test ϕB refers the prepivoted test statistic to a quantile of the uniform (0, 1) distribution. Under regularity conditions, the bootstrap test ϕB has a smaller asymptotic order of error in level than does the asymptotic test ϕA , provided that the asymptotic null distribution of the test statistic does not depend on unknown parameters. In the contrary case, both ϕA and ϕB have the same order of level error. Certain classical refinements to asymptotic tests can be regarded as analytical approximations to the bootstrap test ϕB. These classical results include Welch's estimated t distribution solution to the Behrens-Fisher problem, Bartlett's adjustment to likelihood ratio tests, and Edgeworth expansion corrections to the nonparametric t test for a mean. On the other hand, the bootstrap test ϕB can also be approximated directly by a Monte Carlo algorithm. The prepivoted bootstrap test ϕ B,1 is obtained by prepivoting the test statistic twice before referring the result to a quantile of the uniform (0, 1) distribution. Under regularity conditions, the level error of ϕ B,1 is of smaller asymptotic order than is the level error of either ϕB or ϕA. Analytical and semianalytical approximations sometimes exist for ϕ B,1. In general, a nested double-bootstrap Monte Carlo algorithm yields a satisfactory approximation to ϕ B,1. The possibility of direct nonanalytical implementation is a great practical merit of both ϕB and ϕ B,1.