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Note on distribution free testing for discrete distributions. (English) Zbl 1294.62095
Summary: The paper proposes one-to-one transformation of the vector of components \(\{Y_{in}\}_{i=1}^{m}\) of Pearson’s chi-square statistic,
\[ Y_{in}=\frac{\nu_{in}-np_{i}}{\sqrt{np_{i}}},\quad i=1,\ldots,m, \]
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.

62G10 Nonparametric hypothesis testing
62E20 Asymptotic distribution theory in statistics
62F10 Point estimation
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