Testing Monotone Continuous Distributions on High-Dimensional Real Cubes

  • Michał Adamaszek
  • Artur Czumaj
  • Christian Sohler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6390)

Abstract

We study the task of testing properties of probability distributions and our focus is on understanding the role of continuous distributions in this setting. We consider a scenario in which we have access to independent samples of an unknown distribution \(\mathfrak{D}\) with infinite (perhaps even uncountable) support. Our goal is to test whether \(\mathfrak{D}\) 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 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 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 \(\mathfrak{D}\) on [0,1]n is monotone, then one can test if \(\mathfrak{D}\) is uniform with the sample complexity \(\mathcal{O}(n/\epsilon^2)\). This result is optimal up to a polylogarithmic factor. We also extend the result of Rubinfeld and Servedio (STOC’2005) to test if a distribution \(\mathfrak{D}\) on {0,1,...,k}n is monotone with the sample complexity \(\mathcal{O}(n/\epsilon^2)\).

Keywords

testing distributions 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Michał Adamaszek
    • 1
  • Artur Czumaj
    • 2
  • Christian Sohler
    • 3
  1. 1.Centre for Discrete Mathematics and its Applications (DIMAP) and, Warwick Mathematics InstituteUniversity of WarwickUK
  2. 2.Department of Computer Science and Centre for Discrete Mathematics and, its Applications (DIMAP)University of WarwickUK
  3. 3.Department of Computer ScienceTechnische Universität DortmundGermany

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