On the distinction between the Poisson and Polya distributions when a large number of small samples is available

In the paper we consider the problem of the distinction between the Poisson and Polya distributions when a large number of small samples of different size is available. We use a conditional test when a value of the sufficient statistic is given. For near alternatives the asymptotic power of the test is investigated. We give an expression for the number of observations necessary for the distinction between the hypotheses with the prescribed errors of the first and second type.