The Annals of Statistics

Note on distribution free testing for discrete distributions

Estate Khmaladze

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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.

Article information

Ann. Statist., Volume 41, Number 6 (2013), 2979-2993.

First available in Project Euclid: 1 January 2014

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62D05: Sampling theory, sample surveys 62E20: Asymptotic distribution theory
Secondary: 62E05 62F10: Point estimation

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


Khmaladze, Estate. Note on distribution free testing for discrete distributions. Ann. Statist. 41 (2013), no. 6, 2979--2993. doi:10.1214/13-AOS1176.

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Supplemental materials

  • Supplementary material: Supplement: Distribution free Kolmogorov–Smirnov and Cramér–von Mises tests for power-law distribution. We compare asymptotic behavior of the two classical goodness-of-fit tests based on partial sums of $Y_{in}$’s and their distribution free transformations $Z_{in}$’s and show their power under Zipf’s law and under Karlin–Rouault law as alternatives.