[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/minimum-distance-estimation-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki19\/minimum-distance-estimation-wikipedia\/","headline":"Minimum-distance estimation – Wikipedia","name":"Minimum-distance estimation – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 Method for fitting a statistical model to data Minimum-distance estimation (MDE) is a","datePublished":"2019-03-14","dateModified":"2019-03-14","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki19\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c56d64b5d1d7aef0b45482042a17f140467571b7","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c56d64b5d1d7aef0b45482042a17f140467571b7","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki19\/minimum-distance-estimation-wikipedia\/","about":["Wiki"],"wordCount":2626,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Method for fitting a statistical model to dataMinimum-distance estimation (MDE) is a conceptual method for fitting a statistical model to data, usually the empirical distribution. Often-used estimators such as ordinary least squares can be thought of as special cases of minimum-distance estimation.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4While consistent and asymptotically normal, minimum-distance estimators are generally not statistically efficient when compared to maximum likelihood estimators, because they omit the Jacobian usually present in the likelihood function. This, however, substantially reduces the computational complexity of the optimization problem.Table of ContentsDefinition[edit]Statistics used in estimation[edit]Chi-square criterion[edit]Cram\u00e9r\u2013von Mises criterion[edit]Kolmogorov\u2013Smirnov criterion[edit]Anderson\u2013Darling criterion[edit]Theoretical results[edit]See also[edit]References[edit]Definition[edit]Let        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4X1,\u2026,Xn{displaystyle displaystyle X_{1},ldots ,X_{n}} be an independent and identically distributed (iid) random sample from a population with distribution F(x;\u03b8):\u03b8\u2208\u0398{displaystyle F(x;theta )colon theta in Theta } and \u0398\u2286Rk(k\u22651){displaystyle Theta subseteq mathbb {R} ^{k}(kgeq 1)}.Let Fn(x){displaystyle displaystyle F_{n}(x)} be the empirical distribution function based on the sample.Let \u03b8^{displaystyle {hat {theta }}} be an estimator for \u03b8{displaystyle displaystyle theta }. Then F(x;\u03b8^){displaystyle F(x;{hat {theta }})} is an estimator for F(x;\u03b8){displaystyle displaystyle F(x;theta )}.Let d[\u22c5,\u22c5]{displaystyle d[cdot ,cdot ]} be a functional returning some measure of “distance” between the two arguments. The functional d{displaystyle displaystyle d} is also called the criterion function.If there exists a \u03b8^\u2208\u0398{displaystyle {hat {theta }}in Theta } such that d[F(x;\u03b8^),Fn(x)]=inf{d[F(x;\u03b8),Fn(x)];\u03b8\u2208\u0398}{displaystyle d[F(x;{hat {theta }}),F_{n}(x)]=inf{d[F(x;theta ),F_{n}(x)];theta in Theta }}, then \u03b8^{displaystyle {hat {theta }}} is called the minimum-distance estimate of \u03b8{displaystyle displaystyle theta }.(Drossos & Philippou 1980, p.\u00a0121)Statistics used in estimation[edit]Most theoretical studies of minimum-distance estimation, and most applications, make use of “distance” measures which underlie already-established goodness of fit tests: the test statistic used in one of these tests is used as the distance measure to be minimised. Below are some examples of statistical tests that have been used for minimum-distance estimation.Chi-square criterion[edit]The chi-square test uses as its criterion the sum, over predefined groups, of the squared difference between the increases of the empirical distribution and the estimated distribution, weighted by the increase in the estimate for that group.Cram\u00e9r\u2013von Mises criterion[edit]The Cram\u00e9r\u2013von Mises criterion uses the integral of the squared difference between the empirical and the estimated distribution functions (Parr & Schucany 1980, p.\u00a0616).Kolmogorov\u2013Smirnov criterion[edit]The Kolmogorov\u2013Smirnov test uses the supremum of the absolute difference between the empirical and the estimated distribution functions (Parr & Schucany 1980, p.\u00a0616).Anderson\u2013Darling criterion[edit]The Anderson\u2013Darling test is similar to the Cram\u00e9r\u2013von Mises criterion except that the integral is of a weighted version of the squared difference, where the weighting relates the variance of the empirical distribution function (Parr & Schucany 1980, p.\u00a0616).Theoretical results[edit]The theory of minimum-distance estimation is related to that for the asymptotic distribution of the corresponding statistical goodness of fit tests. Often the cases of the Cram\u00e9r\u2013von Mises criterion, the Kolmogorov\u2013Smirnov test and the Anderson\u2013Darling test are treated simultaneously by treating them as special cases of a more general formulation of a distance measure. Examples of the theoretical results that are available are: consistency of the parameter estimates; the asymptotic covariance matrices of the parameter estimates.See also[edit]References[edit]       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/minimum-distance-estimation-wikipedia\/#breadcrumbitem","name":"Minimum-distance estimation – Wikipedia"}}]}]