Theoretical and numerical aspects of the pattern maximum likelihood estimator, with a view towards symmetric functionals estimation

Abstract: Suppose that an infinite population is partitioned into different species. Given a random sample from the population of species, we are interested in estimating the species richness, which is the number of different species inhabiting a given area. The species richness is a symmetric functional of the probability mass function. A suitable model for estimating the probability mass function is via the pattern maximum likelihood. When the pattern maximum likelihood cannot be found analytically, a sieved version of the pattern maximum likelihood can be used to find a numerical solution to the likelihood problem. The sieved pattern maximum likelihood estimator is consistent and can be calculated numerically using the SAEM algorithm.