Discriminating Between Binomial Distributions

Abstract

Given a set of $k$ random samples, $x_1, x_2, \cdots, x_k$, from a binomial distribution with parameters $p$ and $n$, it is shown that the familiar binomial index of dispersion $z = \frac{\sum^k_1 (x_i - \bar x)^2}{\bar x\big(1 - \frac{\bar x}{n_0}\big)}$ yields an approximate best critical region independent of $p$ for testing the hypothesis $n = n_0$ against the alternative hypothesis $n > n_0$, provided $\bar x$ and $n_0 - \bar x$ are not small. Because of the nature of the test, its optimum properties also apply to testing whether the data came from a binomial population with $n = n_0$ or from a Poisson population.