Note on distribution free testing for discrete distributions
The Annals of Statistics · 2013 · Volume 41 · Issue 6
Abstract
The paper proposes one-to-one transformation of the vector of components $\{Y_{in}\}_{i=1}^{m}$ of Pearson’s chi-square statistic, \[Y_{in}=\frac{\nu_{in}-np_{i}}{\sqrt{np_{i}}},\qquad i=1,\ldots,m,\] 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 $\sum_{I\leq k}Z_{in}$. 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.
Keywords
62D05, 62E05, 62E20, 62F10, Components of chi-square statistics, parametric families of distributions, power-law distributions, projections, unitary transformations, Zipf’s law
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