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

Ruth Heller, Yair Heller, Shachar Kaufman, Barak Brill, Malka Gorfine

Research output: Contribution to journalReview articlepeer-review

54 Scopus citations


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.

Original languageEnglish
JournalJournal of Machine Learning Research
StatePublished - 1 Feb 2016
Externally publishedYes

Bibliographical note

Publisher Copyright:
©2016 Ruth Heller and Yair Heller and Shachar Kaufman and Barak Brill and Malka Gorfine.


R. Heller and B. Brill are supported by Israel Science Foundation grant 2012896. S. Kaufman is supported by a fellowship from the Edmond J. Safra Center for Bioinformatics at Tel Aviv University.

FundersFunder number
Edmond J. Safra Center for Bioinformatics
Israel Science Foundation2012896
Tel Aviv University


    • Bivariate distribution
    • HHG R package
    • Mutual information
    • Nonparametric test
    • Statistical independence
    • Two-sample test


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