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Two-sample testing is a fundamental problem in statistics. While many powerful nonparametric methods exist for both the univariate and multivariate context, it is comparatively less common to see a framework for determining which data features lead to rejection of the null. In this paper, we propose a new nonparametric two-sample test named AUGUST, which incorporates a framework for interpretation while maintaining power comparable to existing methods. AUGUST tests for inequality in distribution up to a predetermined resolution using symmetry statistics from binary expansion. Designed for univariate and low to moderate-dimensional multivariate data, this construction allows us to understand distributional differences as a combination of fundamental orthogonal signals. Asymptotic theory for the test statistic facilitates p-value computation and power analysis, and an efficient algorithm enables computation on large data sets. In empirical studies, we show that our test has power comparable to that of popular existing methods, as well as greater power in some circumstances. We illustrate the interpretability of our method using NBA shooting data.
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This content will become publicly available on July 6, 2027
A Studentized Spherical Harmonics–Based Nonparametric Two-Sample Test for Compositional and Directional Data.
Compositional data analysis has gained increased attention due to the widespread occurrence of simplex-valued data, including microbiome data and financial portfolios. Existing compositional two-sample tests often require $$\log$$-transformations and only detect mean differences, motivating the need for a more general framework without relying on $$\log$$-based methods. There is a close connection between compositional data and directional statistics, and we construct a unified non-parametric two-sample test framework. Our work is based on a studentized energy statistic constructed from spherical harmonics theory over a fixed dimensional underlying space, incorporating U-statistics theory and recent developments of studentization for both compositional and directional data. We establish asymptotic normality for our spherical harmonics based test statistics, thus avoiding the need for permutation tests or bootstrap procedures. Our proposed framework sheds new light on the connections between Non-Euclidean data analysis and classical asymptotic high-dimensional data techniques.
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- Award ID(s):
- 2100729
- PAR ID:
- 10708308
- Publisher / Repository:
- Proceedings of the forty-third International Conference on Machine Learning
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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