The complexity of approximating entropy
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Testing that distributions are close
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Testing Random Variables for Independence and Identity
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Sublinear algorithms for testing monotone and unimodal distributions
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Testing monotone high-dimensional distributions
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Testing k-wise and almost k-wise independence
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Testing symmetric properties of distributions
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Strong Lower Bounds for Approximating Distribution Support Size and the Distinct Elements Problem
SIAM Journal on Computing
Testing monotone continuous distributions on high-dimensional real cubes
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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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 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 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 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. We also extend the result of Rubinfeld and Servedio (STOC'2005) to test if a distribution D on {0, 1, . . . , k}n is monotone with the sample complexity O(n/ε2).