M-estimator
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 are a special case of 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. 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. In many applications, such M-estimators can be thought of as estimating characteristics of the population.
The method of least squares is a prototypical M-estimator, since the estimator is defined as a minimum of the sum of squares of the residuals.
Another popular M-estimator is maximum-likelihood estimation. For a family of probability density functions f parameterized by θ, a maximum likelihood estimator of θ is computed for each set of data by maximizing the likelihood function over the parameter space { θ } . When the observations are independent and identically distributed, a ML-estimate satisfies
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