Note on distribution free testing for discrete distributions

Abstract

The paper proposes one-to-one transformation of the vector of components $\{Y_{in}{\}}_{i=1}^m$ of Pearson's chi-square statistic, \[Y_{in}=\frac{ν_{in}-np_i}{\sqrt{np_i}},\qquad i=1,\ldots,m,\] into another vector $\{Z_{in}{\}}_{i=1}^m$, which, therefore, contains the same "statistical information," but is asymptotically distribution free. Hence any functional/test statistic based on $\{Z_{in}{\}}_{i=1}^m$ is also asymptotically distribution free. Natural examples of such test statistics are traditional goodness-of-fit statistics from partial sums ${\sum }_{I\le k}{Z}_{in}$. The supplement shows how the approach works in the problem of independent interest: the goodness-of-fit testing of power-law distribution with the Zipf law and the Karlin-Rouault law as particular alternatives.

Published in at http://dx.doi.org/10.1214/13-AOS1176 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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