Estimating the proportions of two populations in a mixture using linear maps

We address the problem of estimating the proportions of two statistical populations in a given mixture on the basis of an unlabeled sample of n–dimensional observations on the mixture. Assuming that the expected values of observations on the two populations are known, we show that almost any linear map from Rⁿ to R¹ yields an unbiased consistent estimate of the proportion of one population in a very easy way. We then find that linear map for which the resulting proportion estimate has minimum variance among all estimates so obtained. After deriving a simple expression for the minimum-variance estimate, we discuss practical aspects of obtaining this and related estimates.