On the Distribution of a Symmetric Statistic From a Mixed Population

The density of a symmetric statistic T = g(X 1, X 2, …, Xn ), for a random sample from a mixed population with density f(x) = pf 1(x) + pf 2), is a binomial mixture of the densities of the statist.ics Tk = g(Xk1 , Xk2 , Xkn ), k = 0, 1, … n. where Xki 's are independent with density f 1(x) if i ≤ k and density f 2(x) if i > k. It is shown how to find the distributions of some important symmetric statistics like sample mean, sample variance, and order statistics by using Tk 's. The results are applied to normal and exponential mixtures.