Abstract

This work proposes powerful tests using bootstrap methods for the tail index in the family of distribution functions with nondegenerate right tail. It is of interest to test whether the right tail of a distribution function is the same as or heavier than that of the Pareto distribution with index m₀ for some m₀, vs. alternatively the right tail is lighter. Based on the nonparametric tests of Jureckova and Picek (2001 Jureckova, J., and J. Picek. 2001. A class of tests on the tail index. Extremes 4 (2):165–83. doi:10.1023/A:1013925226836.[Crossref] , [Google Scholar], A class of tests on the tail index, Extreme 4:2, 165–183) that use empirical distribution functions of sample maxima, the proposed bootstrap tests employ two-stage subsamplings: One is subsampling from the sample maxima to be applicable even to the heavy tailed index and the other is subsampling from collections of empirical distributions of the sample maxima. We show the first-order validity of the bootstrap for the nonparametric test and construct bootstrap test statistics. The limiting distribution of the bootstrap test statistics in the null hypothesis is established and the consistency of the bootstrap tests is verified. Applications of the proposed bootstrap tests are given to autoregressive time series models. A simulation study is conducted to see performance of the bootstrap tests and it illustrates that the proposed bootstrap tests outperform the existing ones. Real data of heavy tailed financial stock indices are applied to the proposed bootstrap tests.

摘要

这项工作提出了使用引导方法对具有非退化右尾的分布函数族中的尾部索引进行强大的测试。测试分布函数的右尾是否与索引为m ₀的帕累托分布的右尾相同或比某些m ₀更重，或者右尾更轻是很有趣的。基于 Jureckova 和 Picek 的非参数检验 ( 2001 Jureckova, J.和J. Picek。2001 年。尾部索引的一类测试。极端4 (2)：165 – 83。内政部：10.1023/A：1013925226836。[交叉引用]  , [谷歌学术搜索], 一类使用样本最大值的经验分布函数的尾部指数测试, Extreme 4:2, 165–183), 建议的自举测试采用两阶段二次抽样：一个是从样本最大值进行二次抽样，甚至适用于重尾指数和另一个是从样本最大值的经验分布集合中进行二次抽样。我们展示了非参数检验的 bootstrap 的一阶有效性，并构建了 bootstrap 检验统计数据。建立了原假设下bootstrap检验统计量的极限分布，验证了bootstrap检验的一致性。建议的引导程序测试的应用程序被提供给自回归时间序列模型。进行了模拟研究以查看自举测试的性能，它表明所提出的自举测试优于现有的自举测试。重尾金融股票指数的真实数据被应用于所提出的引导测试。