Estimation of the mixing proportion in a mixture of two normal distributions from simple, rapid measurements

A mixture of two or more normal distributions often provides an adequate model for the distribution of a population consisting of varying proportions of component subpopulations. We consider here the problem of estimating the mixing proportion in a mixture of two normal distributions, the parameters of which can be assumed known. Very large samples may be needed if reasonably precise estimates are to be obtained, thus bringing into consideration the cost or time involved in obtaining large numbers of exact measurements and computing the estimates from them. Simple estimators based on simple, rapidly obtained measurements may then be attractive alternatives provided efficiency losses are not too great. Three such estimators studied here are based on (a) the number of observations less than a fixed point r, (b) the nembers less than s and greater than t, and (c) the sample mean. Optimal choices of the points r, s and t are considered, and the efficiencies of the estimators relative to maximum likelihood estimators (MLE) using the full data are obtained. The simple estimators often perform sufficiently well to make the collection of full data not worthwhile in practice.