## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 18, Number 4 (1947), 556-564.

### 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.

#### Article information

**Source**

Ann. Math. Statist. Volume 18, Number 4 (1947), 556-564.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aoms/1177730346

**Digital Object Identifier**

doi:10.1214/aoms/1177730346

**Mathematical Reviews number (MathSciNet)**

MR23035

**Zentralblatt MATH identifier**

0029.30803

**JSTOR**

links.jstor.org

#### Citation

Hoel, Paul G. Discriminating Between Binomial Distributions. Ann. Math. Statist. 18 (1947), no. 4, 556--564. doi:10.1214/aoms/1177730346. http://projecteuclid.org/euclid.aoms/1177730346.