Testing monotone continuous distributions on high-dimensional real cubes

Authors:
Michat Adamaszek;Artur Czumaj;Christian Sohler
Affiliations:
University of Warwick;University of Warwick;Technische Universität Dortmund
Venue:
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Year:
2010

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Abstract

We study the task of testing properties of probability distributions. We consider a scenario in which we have access to independent samples of an unknown distribution D with infinite (perhaps even uncountable) support. Our goal is to test whether D has a given property or it is ε-far from it (in the statistical distance, with the L1-distance measure). It is not difficult to see that for many natural distributions on infinite or uncountable domains, no testing algorithm can exist and the central objective of our study is to understand if there are any nontrivial distributions that can be efficiently tested. For example, it is easy to see that there is no testing algorithm that tests if a given probability distribution on [0, 1] is uniform. We show however, that if some additional information about the input distribution is known, testing uniform distribution is possible. We extend the recent result about testing uniformity for monotone distributions on Boolean n-dimensional cubes by Rubinfeld and Servedio (STOC'2005) to the case of continuous [0, 1]n cubes. We show that if a distribution D on [0, 1]n is monotone, then one can test if D is uniform with the sample complexity O(n/ε2). This result is optimal up to a polylogarithmic factor.