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December 2013 Note on distribution free testing for discrete distributions
Estate Khmaladze
Ann. Statist. 41(6): 2979-2993 (December 2013). DOI: 10.1214/13-AOS1176


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.


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Estate Khmaladze. "Note on distribution free testing for discrete distributions." Ann. Statist. 41 (6) 2979 - 2993, December 2013.


Published: December 2013
First available in Project Euclid: 1 January 2014

zbMATH: 1294.62095
MathSciNet: MR3161454
Digital Object Identifier: 10.1214/13-AOS1176

Primary: 62D05 , 62E20
Secondary: 62E05 , 62F10

Keywords: Components of chi-square statistics , parametric families of distributions , power-law distributions , projections , unitary transformations , Zipf’s law

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 6 • December 2013
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