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# Statistics and Its Interface

## Volume 16 (2023)

### Number 4

### Metric distributional discrepancy in metric space

Pages: 565 – 578

DOI: https://dx.doi.org/10.4310/22-SII744

#### Authors

#### Abstract

Independence analysis is an indispensable step before regression analysis to find out the essential factors that influence the objects. With many applications in machine Learning, medical Learning and a variety of disciplines, statistical methods of measuring the relationship between random variables have been well studied in vector spaces. However, there are few methods developed to verify the relation between random elements in metric spaces. In this paper, we present a novel index called metric distributional discrepancy (MDD) to measure the dependence between a random element $X$ and a categorical variable $Y$ , which is applicable to the medical image and related variables. The metric distributional discrepancy statistics can be considered as the distance between the conditional distribution of $X$ given each class of $Y$ and the unconditional distribution of $X$. MDD enjoys some significant merits compared to other dependence-measures. For instance, MDD is zero if and only if $X$ and $Y$ are independent. MDD test is a distribution-free test since there is no assumption on the distribution of random elements. Furthermore, MDD test is robust to the data with heavy-tailed distribution and potential outliers. We demonstrate the validity of our theory and the property of the MDD test by several numerical experiments and real data analysis.

#### Keywords

metric distributional discrepancy, random element, metric space, distribution-free test

The authors Wenliang Pan and Yujue Li contributed equally to this work.

Received 13 December 2021

Accepted 8 June 2022

Published 14 April 2023