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Anyone here have any experience with $M$ - estimators and do you think you can give a brief explanation that the problem an $M$ - estimator is trying to solve ? Thanks.

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While your question is interesting, this site is not really geared toward expository essays on broad topics. If you'd like a useful reference, I recommend A. van der Vaart, Asymptotic Statistics, Cambridge University Press, 1998, Chapter 5. –  cardinal Jan 24 '12 at 21:18

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