Robust estimation of finite mixtures of normal distributions

What is it about?

Unlike the MLE (as obtained by the EM algorithm) the L2 estimator has a bounded influence function and performs in a stable manner when there may be contamination. Interestingly the L2 estimator is robust to potential outliers but outperforms the usual robust alternative which is the MLE of a mixture of t-distributions when the data are actually a mixture of normals. Indeed the L2 estimator is consistent and asymptotically normal since the estimator found from the L2 estimating equations is an M-estimator with a bounded and continuous psi function meaning, with some other conditions that are satified,that there is a weakly consitent and Frechet differentiable root to the M-estimating equations.

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Why is it important?

The L2 estimator has all the hallmarks of a robust and efficient estimator when the component distributions are difficult to distinguish, which is exactly when you want to estimate the parameters well. Why should you use a mixture of t-distributions when in fact the data are from a mixture of normals. Using a mixture of t-distributions would give potentially robust estimates but inconsistent estimates when the data are actually normal.

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