Note on distribution free testing for discrete distributions

The paper proposes one-to-one transformation of the vector of components $\{Y_{in}{\}}_{i=1}^m$ of Pearson’s chi-square statistic, $$Y_{in}=\frac{{\nu }_{in}-n{p}_i}{\sqrt{n{p}_i}},i=1,\dots ,m,$$ 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.