Consistent distribution-free $K$-sample and independence tests for univariate random variables

TL;DR

This paper introduces novel, consistent distribution-free tests for univariate independence and $K$-sample problems, utilizing aggregation over partitions of fixed size, with proven efficiency and strong power in simulations and real data.

Contribution

The paper proposes new aggregation-based independence tests that are distribution-free, consistent, and computationally efficient, improving power over existing methods.

Findings

Tests are nearly as powerful as optimal partition-based tests.

Proposed algorithms run in polynomial time.

Regularized tests outperform existing methods in simulations.

Abstract

A popular approach for testing if two univariate random variables are statistically independent consists of partitioning the sample space into bins, and evaluating a test statistic on the binned data. The partition size matters, and the optimal partition size is data dependent. While for detecting simple relationships coarse partitions may be best, for detecting complex relationships a great gain in power can be achieved by considering finer partitions. We suggest novel consistent distribution-free tests that are based on summation or maximization aggregation of scores over all partitions of a fixed size. We show that our test statistics based on summation can serve as good estimators of the mutual information. Moreover, we suggest regularized tests that aggregate over all partition sizes, and prove those are consistent too. We provide polynomial-time algorithms, which are critical for computing the suggested test statistics efficiently. We show that the power of the regularized tests is excellent compared to existing tests, and almost as powerful as the tests based on the optimal (yet unknown in practice) partition size, in simulations as well as on a real data example.