  In this paper, we propose a maximum smoothed likelihood method to estimate
the component density functions of mixture models, in which the mixing
proportions are known and may differ among observations. The proposed estimates
maximize a smoothed log likelihood function and inherit all the important
properties of probability density functions. A majorization-minimization
algorithm is suggested to compute the proposed estimates numerically. In
theory, we show that starting from any initial value, this algorithm increases
the smoothed likelihood function and further leads to estimates that maximize
the smoothed likelihood function. This indicates the convergence of the
algorithm. Furthermore, we theoretically establish the asymptotic convergence
rate of our proposed estimators. An adaptive procedure is suggested to choose
the bandwidths in our estimation procedure. Simulation studies show that the
proposed method is more efficient than the existing method in terms of
integrated squared errors. A real data example is further analyzed.
