**NEW STATISTICAL BOX-TEST AND ITS POWER**

*Igor Semaev and Mehdi M. Hassanzadeh*

**Abstract: **In this paper, statistical testing of $N$ multinomial probabilities is studied and a new box-test, called \emph{Quadratic Box-Test}, is introduced. The statistics of the new test has $\chi^2_s$ limit distribution as $N$ and the number of trials $n$ tend to infinity, where $s$ is a parameter. The well-known empty-box test is a particular case for $s=1$. The proposal is quite different from Pearson's goodness-of-fit test, which requires fixed $N$ while the number of trials is growing, and linear box-tests. We prove that under some conditions on tested distribution the new test's power tends to $1$. That defines a wide region of non-uniform multinomial probabilities distinguishable from the uniform. For moderate $N$ an efficient algorithm to compute the exact values of the first kind error probability is devised.

**Category / Keywords: **Hash-Functions/Statistical Testing, Chi-square Goodness-of-fit Test, Allocation Problem, Empty-Box Test, Linear Box-Test, Quadratic Box-Test, Probability of Errors

**Date: **received 6 Jun 2011, last revised 7 Jul 2011

**Contact author: **igor at ii uib no

**Available format(s): **PDF | BibTeX Citation

**Note: **A number of misprints are fixed, including one in the e-mail address. Some editorial work was applied.

**Version: **20110707:163048 (All versions of this report)

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