A Coincidence-Based Test for Uniformity Given Very Sparsely Sampled Discrete Data

• Published in: IEEE transactions on information theory Vol. 54; no. 10; pp. 4750 - 4755
• Main Author: Paninski, L.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.10.2008; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Computer errors; Computer science; Convex bounds; Data analysis; Data processing; Distributed computing; Distribution channels; Engineering profession; Entropy; Exact sciences and technology; hypothesis testing; Information theory; Information, signal and communications theory; minimax; Minimax techniques; Samples; Statistical analysis; Statistical distributions; Telecommunications and information theory; Testing; Upper bound; Variability; Hypothesis test; Uniform distribution; hypothesis testing; Minimax method; A priori estimation; Discrete data; Convex bounds; minimax
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2008.928987

How many independent samples N do we need from a distribution p to decide that p is epsiv-distant from uniform in an L 1 sense, Sigma i=1 m | p ( i ) - 1/ m | > epsiv? (Here m is the number of bins on which the distribution is supported, and is assumed known a priori .) Somewhat surprisingly, we only need N epsiv 2 Gt m 1/2 to make this decision reliably (this condition is both sufficient and necessary). The test for uniformity introduced here is based on the number of observed ldquocoincidencesrdquo (samples that fall into the same bin), the mean and variance of which may be computed explicitly for the uniform distribution and bounded nonparametrically for any distribution that is known to be epsiv-distant from uniform. Some connections to the classical birthday problem are noted.