  Independent Component Analysis (ICA) models are very popular semiparametric
models in which we observe independent copies of a random vector $X = AS$,
where $A$ is a non-singular matrix and $S$ has independent components. We
propose a new way of estimating the unmixing matrix $W = A^{-1}$ and the
marginal distributions of the components of $S$ using nonparametric maximum
likelihood. Specifically, we study the projection of the empirical distribution
onto the subset of ICA distributions having log-concave marginals. We show
that, from the point of view of estimating the unmixing matrix, it makes no
difference whether or not the log-concavity is correctly specified. The
approach is further justified by both theoretical results and a simulation
study.
