Moment Estimators for the Parameters of a Mixture of Two Binomial Distributions

Abstract

Mixtures of distributions present two types of problems. The first is the problem of identifiability; that is, given that a distribution function $F$ is a probability mixture of distribution functions belonging to some family $\mathscr{F}$, is the mixture unique? This topic has been dealt with quite extensively in recent papers by Robbins [9], Teicher [10] and [11], and others. The second problem is that of estimating the parameters of the individual distribution functions comprising the mixture and the mixing measure. This is clearly possible only if the given mixture is identifiable. K. P. Pearson [5] and C. R. Rao [6] consider the problem of estimation for a mixture of two normal distributions and P. Rider [7 and 8] has recently constructed estimators for mixtures of two of either the exponential, Poisson, binomial, negative binomial or Weibull distributions. In each of these cases the method of maximum likelihood yields highly intractable equations. All of the above estimators have been constructed by the method of moments. In this paper moment estimators will be constructed for a mixture of two binomial distributions, $(n, p_1)$ and $(n, p_2)$. The construction presented here parallels that of Rider [8]. The limiting distributions of the estimators and their asymptotic relative efficiency will be computed. It will be shown that as the binomial parameter $n \rightarrow \infty$ the asymptotic efficiency of the moment estimators tends to unity. Finally, moment estimators will be constructed for the binomial parameters when the mixing parameter $\alpha$ is known. In this case an apparently anomalous result is obtained in that the asymptotic efficiencies of the analogous moment estimators when $\alpha$ is known tend to 0 rather than 1 as $n \rightarrow \infty$. It is pointed out that it is possible, however, to construct moment estimators whose efficiency tends to 1 in this case as well. The estimators so constructed do not depend on the known proportion $\alpha$ when $n \geqq 3$. This suggests a possible explanation of the above anomaly: This fact that the asymptotic efficiencies tend to 0 rather than 1 as $n \rightarrow \infty$ may be due to the failure of these estimators to take into account sample deviations from the true proportions in which the respective populations are present in the mixture.