Tie-break Bootstrap for Nonparametric Rank Statistics

In this paper, we propose a new bootstrap procedure for the empirical copula process. The procedure involves taking pseudo samples of normalized ranks in the same fashion as the classical bootstrap and applying small perturbations to break ties in the normalized ranks. Our procedure is a simple modification of the usual bootstrap based on sampling with replacement, yet it provides noticeable improvement in the finite sample performance. We also discuss how to incorporate our procedure into the time series framework. Since nonparametric rank statistics can be treated as functionals of the empirical copula, our proposal is useful in approximating the distribution of rank statistics in general. As an empirical illustration, we apply our bootstrap procedure to test the null hypotheses of positive quadrant dependence, tail monotonicity, and stochastic monotonicity, using U.S. Census data on spousal incomes in the past fifteen years.

本文针对经验Copula过程（empirical copula process）提出了一种全新的自助法（bootstrap）流程。该流程采用与经典自助法一致的方式，提取标准化秩的伪样本，并通过施加微小扰动以破除标准化秩中的结。本流程仅对基于有放回抽样的常规自助法做出了简单修改，却能在有限样本表现上实现显著提升。本文还探讨了如何将该流程拓展至时间序列框架之中。由于非参数秩统计量可被视为经验Copula的泛函，因此我们的方法总体上可用于近似秩统计量的分布。作为实证示例，我们利用美国过去十五年的夫妻收入人口普查数据，将所提自助法应用于正象限相依、尾部单调性以及随机单调性的原假设检验之中。