On location mixtures with Pólya frequency components

We consider the problem of mixing k random variables where each of the k components results from shifting a common random variable X0 with a certain probability. We show that if X0 admits a density that is a Pólya frequency function with E[X0]=0, then k, a1,…,ak and π1,…,πk are identifiable for any k≥1. We discuss how log-concave maximum likelihood can be used to estimate the mixed and the unknown density f0 when the latter is symmetric.