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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.
For maximum likelihood estimation, the existence of a global maximum of the likelihood function is of the utmost importance. By the extreme value theorem, it suffices that the likelihood function is continuous on a compact parameter space for the maximum likelihood estimator to exist. [7]
Finding a maximum likelihood solution typically requires taking the derivatives of the likelihood function with respect to all the unknown values, the parameters and the latent variables, and simultaneously solving the resulting equations. In statistical models with latent variables, this is usually impossible.
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
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 ...
Maximum likelihood sequence estimation is formally the application of maximum likelihood to this problem. That is, the estimate of { x ( t )} is defined to be a sequence of values which maximize the functional
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
The discrepancy function is a continuous function of the elements of S, the sample covariance matrix, and Σ(θ), the "reproduced" estimate of S obtained by using the parameter estimates and the structural model. In order for "maximum likelihood" to meet the first criterion, it is used in a revised form as the deviance.