A Sequential Nonparametric Two-Sample Test

Un Test Non-paramétrique d'Homogénéité Séquentiel

Alix Lhéritier (1) , Frédéric Cazals (1)

1 ABS - Algorithms, Biology, Structure

Given samples from two distributions, a nonparametric two-sample test aims at determining whether the two distributions are equal or not, based on a test statistic. This statistic may be computed on the whole dataset, or may be computed on a subset of the dataset by a function trained on its complement. We propose a third tier, consisting of functions exploiting a sequential framework to learn the differences while incrementally processing the data. Sequential processing naturally allows optional stopping, which makes our test the first truly sequential nonparametric two-sample test. We show that any sequential predictor can be turned into a sequential two-sample test for which a valid $p$-value can be computed, yielding controlled type I error. We also show that pointwise universal predictors yield consistent tests, which can be built with a nonparametric regressor based on $k$-nearest neighbors in particular. We also show that mixtures and switch distributions can be used to increase power, while keeping consistency.

Mots clés: Hypothesis testing; Sequential prediction; Bayes factor; Nonparametric two-sample test; Regression; Bayesian mixtures; Switch distributions

Domaines: Géométrie algorithmique [cs.CG]; Statistiques [stat]; Machine Learning [stat.ML]

Alix Lhéritier, Frédéric Cazals. A Sequential Nonparametric Two-Sample Test. [Research Report] RR-8704, Inria. 2015, pp.18. ⟨hal-01135608v2⟩