# M-estimator Algorithm

An M-estimator Algorithm is a point estimation algorithm that searches for a minima of sums of functions of the data.

## References

### 2012

• http://en.wikipedia.org/wiki/M-estimation
• In statistics, M-estimators are a broad class of estimators, which are obtained as the minima of sums of functions of the data. Least-squares estimators and many maximum-likelihood estimators are M-estimators. The definition of M-estimators was motivated by robust statistics, which contributed new types of M-estimators. The statistical procedure of evaluating an M-estimator on a data set is called M-estimation.

More generally, an M-estimator may be defined to be a zero of an estimating function.[1][2][3][4][5][6] This estimating function is often the derivative of another statistical function: For example, a maximum-likelihood estimate is often defined to be a zero of the derivative of the likelihood function with respect to the parameter: thus, a maximum-likelihood estimator is often a critical point of the score function.[7] In many applications, such M-estimators can be thought of as estimating characteristics of the population.

1. V. P. Godambe, editor. Estimating functions, volume 7 of Oxford Statistical Science Series. The Clarendon Press Oxford University Press, New York, 1991.
2. Christopher C. Heyde. Quasi-likelihood and its application: A general approach to optimal parameter estimation. Springer Series in Statistics. Springer-Verlag, New York, 1997.
3. D. L. McLeish and Christopher G. Small. The theory and applications of statistical inference functions, volume 44 of Lecture Notes in Statistics. Springer-Verlag, New York, 1988.
4. Parimal Mukhopadhyay. An Introduction to Estimating Functions. Alpha Science International, Ltd, 2004.
5. Christopher G. Small and Jinfang Wang. Numerical methods for nonlinear estimating equations, volume 29 of Oxford Statistical Science Series. The Clarendon Press Oxford University Press, New York, 2003.
6. Sara A. van de Geer. Empirical Processes in M-estimation: Applications of empirical process theory, volume 6 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2000.
7. Ferguson, Thomas S. (1982). "An inconsistent maximum likelihood estimate". Journal of the American Statistical Association 77 (380): 831–834. JSTOR 2287314.