A comparison of the $$L_2$$ L 2 minimum distance estimator and the EM-algorithm when fitting $${\varvec{{k}}}$$ k -component univariate normal mixtures

Brenton R. Clarke, Thomas Davidson, Robert Hammarstrand
  • Statistical Papers, February 2016, Springer Science + Business Media
  • DOI: 10.1007/s00362-016-0747-x

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.

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.

Perspectives

Dr Brenton R. Clarke (Author)
Murdoch University

This research is a confirmation of the theoretical paper by Clarke and Heathcote (1994) which used the well cited paper on M-functionals by Clarke (1983). The paper points to the paper by Clarke and Heathcote for the reason that it is directly involved with mixtures, whereas the paper of Clarke (1983) is broader with many applications of robust functionals applicable in general settings.

The following have contributed to this page: Dr Brenton R. Clarke