  We consider a semiparametric mixture of two univariate density functions
where one of them is known while the weight and the other function are unknown.
Such mixtures have a history of application to the problem of detecting
differentially expressed genes under two or more conditions in microarray data.
Until now, some additional knowledge about the unknown component (e.g. the fact
that it belongs to a location family) has been assumed. As opposed to this
approach, we do not assume any additional structure on the unknown density
function. For this mixture model, we derive a new sufficient identifiability
condition and pinpoint a specific class of distributions describing the unknown
component for which this condition is mostly satisfied. Our approach to
estimation of this model is based on an idea of applying a maximum smoothed
likelihood to what would otherwise have been an ill-posed problem. We introduce
an iterative MM (Majorization-Minimization) algorithm that estimates all of the
model parameters. We establish that the algorithm possesses a descent property
with respect to a log-likelihood objective functional and prove that the
algorithm converges to a minimizer of such an objective functional. Finally, we
also illustrate the performance of our algorithm in a simulation study and
using a real dataset.
