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

For testing goodness of fit it is very popular to use either the χ-statistic or G-statistics (information divergence). Asymptotically both are χ-distributed so an obvious question is which of the two statistics that has a distribution that is closest to the χ-distribution. Surprisingly, when there is only one degree of freedom it seems like the distribution of information divergence is much better approximated by a χ-distribution than the χ-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. The conjecture is proved for Poisson distributions. I. CHOICE OF STATISTIC We consider the problem of testing goodness-of-fit in a discrete setting. Here we shall follow the classic approach to this problem as developed by Pearson, Neyman and Fisher. The question is whether a sample with observation counts (X1, X2, . . . , Xk) has been generated by the distribution Q = (q1, q2, . . . , qk) . For sample size n the counts (X1, X2, . . . , Xk) is assumed to have a multinomial distribution. We introduce the empirical distribution P̂ = ( X1 n , X2 n , . . . , Xk n ) where n denotes the sample size n = X1 +X2 + · · ·+Xk. Often one uses one of the Csiszár [1] f -divergences