Information Divergence is more chi squared distributed than the chi squared statistics

Details Information Divergence is more chi squared distributed than the chi squared statistics Information Divergence is more chi squared distributed than the chi squared statistics

2012-02-06

arxiv.org/abs/1202.1125v1

Mathematics

Statistics Theory

5 pages, submitted to ISIT 2012

Scientific paper

For testing goodness of fit it is very popular to use either the chi squared statistic or G-statistics (information divergence). Asymptotically both are chi squared distributed so an obvious question is which of the two statistics that has a distribution that is closest to the chi squared distribution. Surprisingly the distribution of information divergence is much better approximated by a chi squared distribution than the chi squared statistic. For random variables we introduce a new transformation that transform several important distributions into new random variables that are almost Gaussian. For the binomial distributions and the Poisson distributions we formulate a general conjecture about how close their transform are to the Gaussian, and this conjecture gives much more accurate estimates of the tail probabilities of these distributions than previously published results. The conjecture is proved for Poisson distributions.

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