Maximum L<italic>q</italic>-Likelihood Estimation via the Expectation-Maximization Algorithm: A Robust Estimation of Mixture Models

We introduce a maximum L<italic>q</italic>-likelihood estimation (ML<italic>q</italic>E) of mixture models using our proposed expectation-maximization (EM) algorithm, namely the EM algorithm with L<italic>q</italic>-likelihood (EM-L<italic>q</italic>). Properties of the ML<italic>q</italic>E obtained from the proposed EM-L<italic>q</italic> are studied through simulated mixture model data. Compared with the maximum likelihood estimation (MLE), which is obtained from the EM algorithm, the ML<italic>q</italic>E provides a more robust estimation against outliers for small sample sizes. In particular, we study the performance of the ML<italic>q</italic>E in the context of the gross error model, where the true model of interest is a mixture of two normal distributions, and the contamination component is a third normal distribution with a large variance. A numerical comparison between the ML<italic>q</italic>E and the MLE for this gross error model is presented in terms of Kullback--Leibler (KL) distance and relative efficiency.