M-estimator

In statistics, M-estimators are a broad class of extremum estimators for which the objective function is a sample average.^[1] Both non-linear least squares and maximum likelihood estimation are special cases of M-estimators. The definition of M-estimators was motivated by robust statistics, which contributed new types of M-estimators.^{[citation needed]} However, M-estimators are not inherently robust, as is clear from the fact that they include maximum likelihood estimators, which are in general not robust. 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.^[2]^[3]^[4]^[5]^[6]^[7] This estimating function is often the derivative of another statistical function. For example, a maximum-likelihood estimate is the point where the derivative of the likelihood function with respect to the parameter is zero; thus, a maximum-likelihood estimator is a critical point of the score function.^[8] In many applications, such M-estimators can be thought of as estimating characteristics of the population.

^ Hayashi, Fumio (2000). "Extremum Estimators". Econometrics. Princeton University Press. ISBN 0-691-01018-8.
^ Vidyadhar P. Godambe, editor. Estimating functions, volume 7 of Oxford Statistical Science Series. The Clarendon Press Oxford University Press, New York, 1991.
^ Christopher C. Heyde. Quasi-likelihood and its application: A general approach to optimal parameter estimation. Springer Series in Statistics. Springer-Verlag, New York, 1997.
^ 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.
^ Parimal Mukhopadhyay. An Introduction to Estimating Functions. Alpha Science International, Ltd, 2004.
^ 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.
^ 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.
^ Ferguson, Thomas S. (1982). "An inconsistent maximum likelihood estimate". Journal of the American Statistical Association. 77 (380): 831–834. doi:10.1080/01621459.1982.10477894. JSTOR 2287314.

[1] Hayashi, Fumio (2000). "Extremum Estimators". Econometrics. Princeton University Press. ISBN 0-691-01018-8.

[2] Vidyadhar P. Godambe, editor. Estimating functions, volume 7 of Oxford Statistical Science Series. The Clarendon Press Oxford University Press, New York, 1991.

[3] Christopher C. Heyde. Quasi-likelihood and its application: A general approach to optimal parameter estimation. Springer Series in Statistics. Springer-Verlag, New York, 1997.

[4] 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.

[5] Parimal Mukhopadhyay. An Introduction to Estimating Functions. Alpha Science International, Ltd, 2004.

[6] 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.

[7] 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.

[8] Ferguson, Thomas S. (1982). "An inconsistent maximum likelihood estimate". Journal of the American Statistical Association. 77 (380): 831–834. doi:10.1080/01621459.1982.10477894. JSTOR 2287314.

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