On adaptive confidence sets for the Wasserstein distances

IF 1.5 2区 数学 Q2 STATISTICS & PROBABILITY
Bernoulli Pub Date : 2021-11-16 DOI:10.3150/22-bej1535
N. Deo, Thibault Randrianarisoa
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引用次数: 1

Abstract

In the density estimation model, we investigate the problem of constructing adaptive honest confidence sets with radius measured in Wasserstein distance $W_p$, $p\geq1$, and for densities with unknown regularity measured on a Besov scale. As sampling domains, we focus on the $d-$dimensional torus $\mathbb{T}^d$, in which case $1\leq p\leq 2$, and $\mathbb{R}^d$, for which $p=1$. We identify necessary and sufficient conditions for the existence of adaptive confidence sets with diameters of the order of the regularity-dependent $W_p$-minimax estimation rate. Interestingly, it appears that the possibility of such adaptation of the diameter depends on the dimension of the underlying space. In low dimensions, $d\leq 4$, adaptation to any regularity is possible. In higher dimensions, adaptation is possible if and only if the underlying regularities belong to some interval of width at least $d/(d-4)$. This contrasts with the usual $L_p-$theory where, independently of the dimension, adaptation requires regularities to lie in a small fixed-width window. For configurations allowing these adaptive sets to exist, we explicitly construct confidence regions via the method of risk estimation, centred at adaptive estimators. Those are the first results in a statistical approach to adaptive uncertainty quantification with Wasserstein distances. Our analysis and methods extend more globally to weak losses such as Sobolev norm distances with negative smoothness indices.
关于Wasserstein距离的自适应置信集
在密度估计模型中,我们研究了在Wasserstein距离$W_p$, $p\geq1$中测量半径和在Besov尺度上测量具有未知规律性的密度的自适应诚实置信集的构建问题。作为采样域,我们将重点放在$d-$维度环面$\mathbb{T}^d$上,在这种情况下是$1\leq p\leq 2$,在$\mathbb{R}^d$上是$p=1$。我们确定了直径为正则相关$W_p$ -minimax估计率阶的自适应置信集存在的充分必要条件。有趣的是,这种直径调整的可能性似乎取决于底层空间的大小。在低维度,$d\leq 4$,适应任何规则是可能的。在更高的维度中,当且仅当潜在的规律属于至少$d/(d-4)$的宽度区间时,适应是可能的。这与通常的$L_p-$理论形成了对比,在理论中,适应需要规律存在于一个固定宽度的小窗口中,而不依赖于维度。对于允许这些自适应集存在的配置,我们通过以自适应估计量为中心的风险估计方法显式地构建置信区域。这是采用统计方法对Wasserstein距离进行自适应不确定性量化的第一个结果。我们的分析和方法更广泛地扩展到具有负平滑指数的Sobolev范数距离等弱损失。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Bernoulli
Bernoulli 数学-统计学与概率论
CiteScore
3.40
自引率
0.00%
发文量
116
审稿时长
6-12 weeks
期刊介绍: BERNOULLI is the journal of the Bernoulli Society for Mathematical Statistics and Probability, issued four times per year. The journal provides a comprehensive account of important developments in the fields of statistics and probability, offering an international forum for both theoretical and applied work. BERNOULLI will publish: Papers containing original and significant research contributions: with background, mathematical derivation and discussion of the results in suitable detail and, where appropriate, with discussion of interesting applications in relation to the methodology proposed. Papers of the following two types will also be considered for publication, provided they are judged to enhance the dissemination of research: Review papers which provide an integrated critical survey of some area of probability and statistics and discuss important recent developments. Scholarly written papers on some historical significant aspect of statistics and probability.
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