A Sequential Non-Parametric Multivariate Two-Sample Test

• Published in: IEEE transactions on information theory Vol. 64; no. 5; pp. 3361 - 3370
• Main Authors: Lheritier, Alix, Cazals, Frederic
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.05.2018; The Institute of Electrical and Electronics Engineers, Inc. (IEEE); Institute of Electrical and Electronics Engineers
• Subjects: Bayes factor; Bayesian analysis; Bayesian mixtures; Computation; Computational Geometry; Computer Science; Datasets; Hypothesis testing; Information theory; Multiplexing; non-parametric two-sample test; Nonparametric statistics; Probabilistic logic; Random variables; regression; sequential prediction; Sociology; State of the art; Statistical analysis; Statistical methods; Statistics; switch distributions; Switches; Testing; universal distributions
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2018.2800658

Given samples from two distributions, a non-parametric two-sample test aims at determining whether the two distributions are equal or not, based on a test statistic. Classically, this statistic is computed on the whole data set, or is computed on a subset of the data set by a function trained on its complement. We consider methods in a third tier, so as to deal with large (possibly infinite) data sets, and to automatically determine the most relevant scales to work at, making two contributions. First, we develop a generic sequential non-parametric testing framework, in which the sample size need not be fixed in advance. This makes our test a truly sequential non-parametric multivariate two-sample test. Under information theoretic conditions qualifying the difference between the tested distributions, consistency of the two-sample test is established. Second, we instantiate our framework using nearest neighbor regressors, and show how the power of the resulting two-sample test can be improved using Bayesian mixtures and switch distributions. This combination of techniques yields automatic scale selection, and experiments performed on challenging data sets show that our sequential tests exhibit comparable performances to those of state-of-the-art non-sequential tests.