The limit joint distributions of statistics of the NIST tests and their generalizations

Abstract

Consider the problem of testing the hypothesis H 0 against the alternative H1, where the hypothesis H0 is that the tested sequence consists of independent random variables with a given polynomial distribution, and the alternative hypothesis H1 corresponds to a scheme of trials in which the distribution of the tested sequence approaches its distribution under H0. To solve this problem, we consider four types of statistics, that are generalizations of statistics of tests of the NIST package and other packages. We present the limiting joint distribution of statistics of these 4 types and the estimate of the rate of convergence to this limiting distribution (analogue of Berry-Esseen inequality). We also present necessary and sufficient conditions for asymptotic independence of the statistics under consideration