Ball Divergence: Nonparametric two sample test

Abstract

In this paper, we first introduce Ball Divergence, a novel measure of the difference between two probability measures in separable Banach spaces, and show that the Ball Divergence of two probability measures is zero if and only if these two probability measures are identical without any moment assumption. Using Ball Divergence, we present a metric rank test procedure to detect the equality of distribution measures underlying independent samples. It is therefore robust to outliers or heavy-tail data. We show that this multivariate two sample test statistic is consistent with the Ball Divergence, and it converges to a mixture of χ2 distributions under the null hypothesis and a normal distribution under the alternative hypothesis. Importantly, we prove its consistency against a general alternative hypothesis. Moreover, this result does not depend on the ratio of the two imbalanced sample sizes, ensuring that can be applied to imbalanced data. Numerical studies confirm that our test is superior to several existing tests in terms of Type I error and power. We conclude our paper with two applications of our method: one is for virtual screening in drug development process and the other is for genome wide expression analysis in hormone replacement therapy.

A robust and nonparametric two-sample test in high dimensions

“…Examples include Baringhaus and Franz (2004), Rosenbaum (2005), and , which are referred to as BF, R, and BG, respectively. We also include Pan et al (2018) in our numerical comparison, which belongs to the third class in which no moment condition or tuning parameter is required. We refer to this test as PTWZ, and our test as QXZ.…”

“…INTRODUCTION4 outliers or extreme values. Pan et al (2018) introduced ball divergence to measure the differences between the empirical probability density functions of the two random samples. It requires no moment condition and is free of tuning parameters.…”

Ball: An R Package for Detecting Distribution Difference and Association in Metric Spaces

“…Recently, two novel concepts, ball divergence (Pan et al 2018) and ball covariance (Pan et al 2020), are proposed to measure the discrepancy between two distributions and the dependence between two random objects in metric spaces, respectively. Ball divergence (BD) enjoys a remarkable property, homogeneity-zero equivalence, and ball covariance (BCOV) holds another brilliant property, independence-zero equivalence.…”

“…The BD and BCOV statistics, as the empirical versions of BD and BCOV, can tackle the two-sample test and test of independence problems, which are special cases of the K-sample test and test of mutual independence problems. The BD and BCOV statistics are both robust rank statistics, and the test procedures based on them are consistent against any general alternative hypothesis without distribution or moment assumptions (Pan et al 2018(Pan et al , 2020. Besides, the BD statistic is proved to cope well with imbalanced data, and the BCOV statistic can be standardized to the ball correlation statistic to extract important features from ultra-high dimensional data (Pan, Wang, Xiao, and Zhu 2019).…”

Distance‐based analysis of variance for brain connectivity

“…More recently, other manifoldbased testing procedures that leverage distance between the spaces in which the observed data reside have been proposed. These have based inference on the bootstrap (Pan et al, 2017) or asymptotic approximations for particular ball-based divergences (Pan et al, 2018).…”