Geodesic
On a randomized approach to the construction of tests for the equiprobability of outcomes of a multinomial scheme
Diskretnaya Matematika, Tome 9 (1997) no. 3, pp. 43-51.

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We consider the most powerful test and the $\chi^2$ test for verifying equiprobability of the outcomes of multinomial scheme in the case where the alternative distribution is obtained by averaging all non-equiprobable multinomial distributions with $N$ outcomes over the symmetric $(N-1)$-dimensional Dirichlet distribution. Limit theorems on the convergence of the distributions of the test statistics under both of the hypotheses are proved. 
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     author = {A. S. Ambrosimov and A. N. Timashev},
     title = {On a randomized approach to the construction of tests for the equiprobability of outcomes of a multinomial scheme},
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A. S. Ambrosimov; A. N. Timashev. On a randomized approach to the construction of tests for the equiprobability of outcomes of a multinomial scheme. Diskretnaya Matematika, Tome 9 (1997) no. 3, pp. 43-51. http://geodesic.mathdoc.fr/item/DM_1997_9_3_a3/