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In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model , the observed data is most probable.
In statistics, the restricted (or residual, or reduced) maximum likelihood (REML) approach is a particular form of maximum likelihood estimation that does not base estimates on a maximum likelihood fit of all the information, but instead uses a likelihood function calculated from a transformed set of data, so that nuisance parameters have no effect.
But for practical purposes it is more convenient to work with the log-likelihood function in maximum likelihood estimation, in particular since most common probability distributions—notably the exponential family—are only logarithmically concave, [34] [35] and concavity of the objective function plays a key role in the maximization.
IRLS is used to find the maximum likelihood estimates of a generalized linear model, and in robust regression to find an M-estimator, as a way of mitigating the influence of outliers in an otherwise normally-distributed data set, for example, by minimizing the least absolute errors rather than the least square errors.
They proposed an iteratively reweighted least squares method for maximum likelihood estimation (MLE) of the model parameters. MLE remains popular and is the default method on many statistical computing packages. Other approaches, including Bayesian regression and least squares fitting to variance stabilized responses, have been developed.
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.
Fisher introduced the concept of likelihood and its maximization as a criterion for estimating parameters. Fisher's approach emphasized the concept of sufficiency and the maximum likelihood estimation (MLE). Likelihoodism can be seen as an extension of Fisherian statistics, refining and expanding the use of likelihood in statistical inference.
Scoring algorithm, also known as Fisher's scoring, [1] is a form of Newton's method used in statistics to solve maximum likelihood equations numerically, named after Ronald Fisher. Sketch of derivation