Editing Maximum likelihood estimation (section)

{{Short description|Method of estimating the parameters of a statistical model, given observations}}
{{About|the statistical techniques|computer data storage|partial-response maximum-likelihood}}

In [[statistics]], '''maximum likelihood estimation''' ('''MLE''') is a method of [[estimation theory|estimating]] the [[Statistical parameter|parameters]] of an assumed [[probability distribution]], given some observed data. This is achieved by [[Mathematical optimization|maximizing]] a [[likelihood function]] so that, under the assumed [[statistical model]], the [[Realization (probability)|observed data]] is most probable. The [[point estimate|point]] in the [[parameter space]] that maximizes the likelihood function is called the maximum likelihood estimate.<ref>{{cite book |last=Rossi |first=Richard J. |title=Mathematical Statistics: An Introduction to Likelihood Based Inference |location=New York |publisher=John Wiley & Sons |year=2018 |isbn=978-1-118-77104-4 |page=227 }}</ref> The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of [[statistical inference]].<ref>{{cite book |first1=David F. |last1=Hendry |author-link=David Forbes Hendry |first2=Bent |last2=Nielsen |title=Econometric Modeling: A Likelihood Approach |location=Princeton |publisher=Princeton University Press |year=2007 |isbn=978-0-691-13128-3 }}</ref><ref>{{cite book |first1=Raymond L. |last1=Chambers |first2=David G. |last2=Steel |first3=Suojin |last3=Wang |first4=Alan |last4=Welsh |title=Maximum Likelihood Estimation for Sample Surveys |location=Boca Raton |publisher=CRC Press |year=2012 |isbn=978-1-58488-632-7 }}</ref><ref>{{cite book |first1=Michael Don |last1=Ward |author-link=Michael D. Ward |first2=John S. |last2=Ahlquist |title=Maximum Likelihood for Social Science: Strategies for Analysis |location=New York |publisher=Cambridge University Press |year=2018 |isbn=978-1-107-18582-1 }}</ref>

If the likelihood function is [[Differentiable function|differentiable]], the [[derivative test]] for finding maxima can be applied. In some cases, the first-order conditions of the likelihood function can be solved analytically; for instance, the [[ordinary least squares]] estimator for a [[linear regression]] model maximizes the likelihood when the random errors are assumed to have [[Normal distribution|normal]] distributions with the same variance.<ref>{{cite book |last1=Press |first1=W.H. |last2=Flannery |first2=B.P. |last3=Teukolsky |first3=S.A. |last4=Vetterling |first4=W.T. |chapter=Least Squares as a Maximum Likelihood Estimator |title=Numerical Recipes in FORTRAN: The Art of Scientific Computing |edition=2nd |location=Cambridge |publisher=Cambridge University Press |pages=651–655 |year=1992 |isbn=0-521-43064-X |chapter-url=https://books.google.com/books?id=gn_4mpdN9WkC&pg=PA651 }}</ref>

From the perspective of [[Bayesian inference]], MLE is generally equivalent to [[maximum a posteriori estimation|maximum a posteriori (MAP) estimation]] with a [[Prior probability|prior distribution]] that is [[uniform distribution (continuous)|uniform]] in the region of interest. In [[frequentist inference]], MLE is a special case of an [[extremum estimator]], with the objective function being the likelihood.