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In this paper, we propose a maximum smoothed likelihood method to estimatethe component density functions of mixture models, in which the mixingproportions are known and may differ among observations. The proposed estimatesmaximize a smoothed log likelihood function and inherit all the importantproperties of probability density functions. A majorization-minimizationalgorithm is suggested to compute the proposed estimates numerically. Intheory, we show that starting from any initial value, this algorithm increasesthe smoothed likelihood function and further leads to estimates that maximizethe smoothed likelihood function. This indicates the convergence of thealgorithm. Furthermore, we theoretically establish the asymptotic convergencerate of our proposed estimators. An adaptive procedure is suggested to choosethe bandwidths in our estimation procedure. Simulation studies show that theproposed method is more efficient than the existing method in terms ofintegrated squared errors. A real data example is further analyzed.

Maximum Smoothed Likelihood Component Density Estimation in Mixture Models with Known Mixing Proportions