  We consider a two-component mixture model with one known component. We
develop methods for estimating the mixing proportion and the unknown
distribution nonparametrically, given i.i.d.~data from the mixture model, using
ideas from shape restricted function estimation. We establish the consistency
of our estimators. We find the rate of convergence and asymptotic limit of the
estimator for the mixing proportion. Completely automated distribution-free
honest finite sample lower confidence bounds are developed for the mixing
proportion. Connection to the problem of multiple testing is discussed. The
identifiability of the model, and the estimation of the density of the unknown
distribution are also addressed. We compare the proposed estimators, which are
easily implementable, with some of the existing procedures through simulation
studies and analyse two data sets, one arising from an application in astronomy
and the other from a microarray experiment.
