Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach
Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach
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DOI：
10.1214/20-aos1996
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发表时间：
2021-04
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通讯作者：
M. Hallin;E. Barrio;J. A. Cuesta-Albertos;C. Matrán
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文献类型：
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作者：
M. Hallin;E. Barrio;J. A. Cuesta-Albertos;C. Matrán
与实线不同，当d≥2时，实空间Rd不是正则有序的。因此，像分位数和分布函数这样的基本单变量概念，以及它们的经验对应物，包括秩和符号，通常不能扩展到多变量环境中。半个多世纪以来，缓解这种缺乏规范排序的问题一直是一个悬而未决的问题，产生了大量的文献，并推动了统计深度和基于耦合的方法的发展。我们表明，与文献中提出的许多定义不同，Chernozhukov等人（2017）引入的基于测度运输的秩具有使单变量秩成为半参数推理成功工具的所有属性。与这些排名相关，我们提出了一个新的多元分布和分位数函数的中心向外定义，以及它们的经验对应物，为此我们建立了Glivenko-Cantelli结果。我们的方法基于McCann(1995)，与Chernozhukov等人（2017）的结果不同，我们的结果不需要任何时刻假设。所得到的秩和符号在Basu（1959）的意义上被证明是严格无分布的，本质上是最大的辅助，在涉及未指定密度噪声的半参数模型中，可以被解释为半参数效率的有限样本形式。虽然构成了样本的充分总结，但经验中心向外分布函数仅在观测值处定义。提供了对整个d维空间的连续扩展，在保留基本单调性和Glivenko-Cantelli特征的同时，产生光滑的经验分位数轮廓和符号曲线。对所得的经验分位数轮廓进行了数值研究。MSC 2010学科分类：初级62G30；二次62 b05。
Unlike the real line, the real space Rd, for d ≥ 2, is not canonically ordered. As a consequence, such fundamental univariate concepts as quantile and distribution functions, and their empirical counterparts, involving ranks and signs, do not canonically extend to the multivariate context. Palliating that lack of a canonical ordering has been an open problem for more than half a century, generating an abundant literature and motivating, among others, the development of statistical depth and copulabased methods. We show that, unlike the many definitions proposed in the literature, the measure transportation-based ranks introduced in Chernozhukov et al. (2017) enjoy all the properties that make univariate ranks a successful tool for semiparametric inference. Related with those ranks, we propose a new center-outward definition of multivariate distribution and quantile functions, along with their empirical counterparts, for which we establish a Glivenko-Cantelli result. Our approach is based on McCann (1995) and our results, unlike those of Chernozhukov et al. (2017), do not require any moment assumptions. The resulting ranks and signs are shown to be strictly distribution-free and essentially maximal ancillary in the sense of Basu (1959) which, in semiparametric models involving noise with unspecified density, can be interpreted as a finite-sample form of semiparametric efficiency. Although constituting a sufficient summary of the sample, empirical center-outward distribution functions are defined at observed values only. A continuous extension to the entire d-dimensional space, yielding smooth empirical quantile contours and sign curves while preserving the essential monotonicity and Glivenko-Cantelli features, is provided. A numerical study of the resulting empirical quantile contours is conducted. MSC 2010 subject classifications: Primary 62G30; secondary 62B05.