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We consider a semiparametric mixture of two univariate density functionswhere 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 detectingdifferentially expressed genes under two or more conditions in microarray data.Until now, some additional knowledge about the unknown component (e.g. the factthat it belongs to a location family) has been assumed. As opposed to thisapproach, we do not assume any additional structure on the unknown densityfunction. For this mixture model, we derive a new sufficient identifiabilitycondition and pinpoint a specific class of distributions describing the unknowncomponent for which this condition is mostly satisfied. Our approach toestimation of this model is based on an idea of applying a maximum smoothedlikelihood to what would otherwise have been an ill-posed problem. We introducean iterative MM (Majorization-Minimization) algorithm that estimates all of themodel parameters. We establish that the algorithm possesses a descent propertywith respect to a log-likelihood objective functional and prove that thealgorithm converges to a minimizer of such an objective functional. Finally, wealso illustrate the performance of our algorithm in a simulation study andusing a real dataset.

A maximum smoothed likelihood based estimation for two component semiparametric density mixtures with a known component