高维距离相关推断的渐近分布。

IF 3.2 1区 数学 Q1 STATISTICS & PROBABILITY
Annals of Statistics Pub Date : 2021-08-01 Epub Date: 2021-09-29 DOI:10.1214/20-aos2024
Lan Gao, Yingying Fan, Jinchi Lv, Qi-Man Shao
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引用次数: 0

摘要

距离相关性已成为检测一对潜在高维随机向量之间非线性依赖关系的一种日益流行的工具。大多数现有研究都探讨了距离相关在两个随机向量之间独立性的零假设下的渐近分布,即只有样本大小或维度发生偏离时的渐近分布。然而,在更现实的情况下,当样本大小和维度都在全范围内发散时,其渐近零分布在很大程度上仍未得到充分发展。在本文中,我们填补了这一空白,并在一些温和的正则条件和零假设下,建立了基于高维度偏差校正距离相关性的重标检验统计量的中心极限定理和相关收敛率。我们的新理论结果揭示了高维距离相关推断的一个有趣的维度祝福现象,即正态逼近的准确性会随着维度的增加而增加。此外,我们还提供了关于依赖性替代假设下幂次分析的一般理论,并进一步证明了在某类替代假设下,重标度距离相关在中等高维条件下捕捉纯非线性依赖性的能力。通过几个模拟实例和一个区块链应用,说明了重标度统计量的理论结果和有限样本性能。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

ASYMPTOTIC DISTRIBUTIONS OF HIGH-DIMENSIONAL DISTANCE CORRELATION INFERENCE.

ASYMPTOTIC DISTRIBUTIONS OF HIGH-DIMENSIONAL DISTANCE CORRELATION INFERENCE.

ASYMPTOTIC DISTRIBUTIONS OF HIGH-DIMENSIONAL DISTANCE CORRELATION INFERENCE.

ASYMPTOTIC DISTRIBUTIONS OF HIGH-DIMENSIONAL DISTANCE CORRELATION INFERENCE.

Distance correlation has become an increasingly popular tool for detecting the nonlinear dependence between a pair of potentially high-dimensional random vectors. Most existing works have explored its asymptotic distributions under the null hypothesis of independence between the two random vectors when only the sample size or the dimensionality diverges. Yet its asymptotic null distribution for the more realistic setting when both sample size and dimensionality diverge in the full range remains largely underdeveloped. In this paper, we fill such a gap and develop central limit theorems and associated rates of convergence for a rescaled test statistic based on the bias-corrected distance correlation in high dimensions under some mild regularity conditions and the null hypothesis. Our new theoretical results reveal an interesting phenomenon of blessing of dimensionality for high-dimensional distance correlation inference in the sense that the accuracy of normal approximation can increase with dimensionality. Moreover, we provide a general theory on the power analysis under the alternative hypothesis of dependence, and further justify the capability of the rescaled distance correlation in capturing the pure nonlinear dependency under moderately high dimensionality for a certain type of alternative hypothesis. The theoretical results and finite-sample performance of the rescaled statistic are illustrated with several simulation examples and a blockchain application.

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来源期刊
Annals of Statistics
Annals of Statistics 数学-统计学与概率论
CiteScore
9.30
自引率
8.90%
发文量
119
审稿时长
6-12 weeks
期刊介绍: The Annals of Statistics aim to publish research papers of highest quality reflecting the many facets of contemporary statistics. Primary emphasis is placed on importance and originality, not on formalism. The journal aims to cover all areas of statistics, especially mathematical statistics and applied & interdisciplinary statistics. Of course many of the best papers will touch on more than one of these general areas, because the discipline of statistics has deep roots in mathematics, and in substantive scientific fields.
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