# M-estimation

July 11, 2016 β February 17, 2022

Loosely, estimating a quantity by choosing it to be the extremum of a function, or, if itβs well-behaved enough, a zero of its derivative.

Popular with machine learning, where loss-function based methods are ubiquitous. In statistics we see this famously in maximum likelihood estimation and robust estimation, and least squares loss, for which M-estimation provides a unifying formalism with a convenient large sample asymptotic theory.

π Discuss influence function motivation.

## 1 Implied density functions

Common loss function imply a density considered as a maximum likelihood estimation problem.

I assume they did not invent this idea, but Davison and Ortiz (2019) points out that if you have a least-squares-compatible model, usually it can generalise it to any elliptical density, which includes Huber losses and many robust ones as special cases.

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## 3 Fitting

Discuss representation (and implementation) in terms of weight functions for least-squares loss.

## 4 GM-estimators

Mallows, Schweppe etc.

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## 5 References

Barndorff-Nielsen. 1983. Biometrika.
BΓΌhlmann. 2014. βRobust Statistics.β In Selected Works of Peter J. Bickel. Selected Works in Probability and Statistics 13.
DasGupta. 2008. Asymptotic Theory of Statistics and Probability. Springer Texts in Statistics.
Davison, and Ortiz. 2019. arXiv:1910.14139 [Cs].
Donoho, and Montanari. 2013. arXiv:1310.7320 [Cs, Math, Stat].
Hampel. 1974. Journal of the American Statistical Association.
Hampel, Ronchetti, Rousseeuw, et al. 2011. Robust Statistics: The Approach Based on Influence Functions.
Huber. 1964. The Annals of Mathematical Statistics.
Kandasamy, Krishnamurthy, Poczos, et al. 2014. arXiv:1411.4342 [Stat].
KΓΌmmel. 1982. Energy.
Markatou, Marianthi, Karlis, and Ding. 2021. Annual Review of Statistics and Its Application.
Markatou, M., and Ronchetti. 1997. In Handbook of Statistics. Robust Inference.
Maronna. 1976. The Annals of Statistics.
Mondal, and Percival. 2010. Annals of the Institute of Statistical Mathematics.
Ortiz, Evans, and Davison. 2021. arXiv:2107.02308 [Cs].
Ronchetti, Elvezio. 1997. Journal of Statistical Planning and Inference, Robust Statistics and Data Analysis, Part I,.
Ronchetti, E. 2000. In Data Segmentation and Model Selection for Computer Vision.
Tharmaratnam, and Claeskens. 2013. Statistics.
van de Geer. 2014. In arXiv:1403.7023 [Math, Stat].
Yang, Gallagher, and McMahan. 2019. Communications in Statistics - Theory and Methods.