Estimators of the Probability of the Zero Class in Poisson and Certain Related Populations

Abstract

Two estimators of the probability of falling into the zero class are compared, for a family of populations related to Poisson populations. The first estimator, $\epsilon_1$, is based on the observed proportion in the zero class; the second, $\epsilon_2$, would be the maximum likelihood estimator if the underlying distribution were Poisson. From a practical point of view each estimator possesses its own peculiar advantages. $\epsilon_1$ has the advantage that the detailed distribution among the non-zero classes need not be examined. $\epsilon_2$ has the advantage that only the mean of the observations is needed, the distribution among the various classes not being required. The relative importance of these advantages will naturally vary according to the situations in which the estimators are to be used. An arbitrary measure of relative accuracy, the mean square error ratio, is used. On this basis $\epsilon_2$ is superior to $\epsilon_1$ for all sample sizes (greater than one) if the population distribution is Poisson. Provided the sample size is not too large $\epsilon_2$ may still be superior to $\epsilon_1$ when the population distribution deviates to a moderate extent from Poisson form. A third estimator $\epsilon_3$, which is a modification of $\epsilon_2$ and is unbiased, provided the population is Poisson, may be preferred to $\epsilon_2$ unless $p$ exceeds about 0.45. Its properties vis-a-vis $\epsilon_1$ probably differ little from those of $\epsilon_2$.