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'''Minimum distance estimation (MDE)''' is a statistical method for fitting a mathematical model to data, usually the [[Empirical distribution function|empirical distribution]].
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==Definition==
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Let <math>\displaystyle X_1,\ldots,X_n</math> be an [[Independent and identically-distributed random variables|independent and identically distributed]] (iid) [[Random variable|random]] [[Random sample|sample]] from a [[Statistical population|population]] with [[Cumulative distribution function|distribution]] <math>F(x;\theta)\colon \theta\in\Theta</math> and <math>\Theta\subseteq\mathbb{R}^k (k\geq 1)</math>.
 
Let <math>\displaystyle F_n(x)</math> be the [[empirical distribution function]] based on the sample.
 
Let <math>\hat{\theta}</math> be an [[estimator]] for <math>\displaystyle \theta</math>. Then <math>F(x;\hat{\theta})</math> is an estimator for <math>\displaystyle F(x;\theta)</math>.
 
Let <math>d[\cdot,\cdot]</math> be a [[Functional (mathematics)|functional]] returning some measure of [[Distance#General_case|"distance"]] between the two [[Dependent and independent variables#Mathematics|arguments]]. The functional <math>\displaystyle d</math> is also called the criterion function.
 
If there exists a <math>\hat{\theta}\in\Theta</math> such that <math>d[F(x;\hat{\theta}),F_n(x)]=\inf\{d[F(x;\theta),F_n(x)]; \theta\in\Theta\}</math>, then <math>\hat{\theta}</math> is called the '''minimum distance estimate''' of <math>\displaystyle \theta</math>.
 
{{harv|Drossos|Philippou|1980|p=121}}
 
==Statistics used in estimation==
 
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===
 
The [[Pearson's chi-square test|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.
 
===Cramér–von Mises criterion===
 
The [[Cramér–von Mises criterion]] uses the integral of the squared difference between the empirical and the estimated distribution functions {{harv|Parr|Schucany|1980|p=616}}.
 
===Kolmogorov–Smirnov criterion===
 
The [[Kolmogorov–Smirnov test]] uses the [[supremum]] of the [[absolute difference]] between the empirical and the estimated distribution functions {{harv|Parr|Schucany|1980|p=616}}.
 
===Anderson–Darling criterion===
 
The [[Anderson–Darling test]] is similar to the Cramér–von 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 {{harv|Parr|Schucany|1980|p=616}}.
 
==Theoretical results==
 
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 [[Cramér–von Mises criterion]], the [[Kolmogorov–Smirnov test]] and the [[Anderson–Darling 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 (statistics)|consistency]] of the parameter estimates; the asymptotic covariance matrices of the parameter estimates.
 
==See also==
*[[Maximum likelihood estimation]]
*[[Maximum spacing estimation]]
 
==References==
{{reflist}}
{{refbegin}}
*{{cite journal
| last1 = Boos
| first1 = Dennis D.
| year = 1982
| title = Minimum anderson-darling estimation
| journal = Communications in Statistics - Theory and Methods
| volume = 11
| issue = 24
| pages = 2747–2774
| publisher = [[Taylor & Francis]]
| issn = 0361-0926
| doi = 10.1080/03610928208828420
}}
*{{cite journal
|last= Blyth
|first= Colin R.
|date=June 1970
|title= On the Inference and Decision Models of Statistics
|journal= The Annals of Mathematical Statistics
|volume= 41
|issue= 3
|pages= 1034–1058
|publisher= [[Institute of Mathematical Statistics]]
|issn= 0020-3157
|doi= 10.1214/aoms/1177696980
|url= http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aoms/1177696980
|format= PDF
|accessdate= 2008-09-24
}}
*{{cite journal
|last1= Drossos
|first1= Constantine A.
|last2= Philippou
|first2= Andreas N.
|date=December 1980
|title= A Note on Minimum Distance Estimates
|journal= Annals of the Institute of Statistical Mathematics
|volume= 32
|issue= 1
|pages= 121–123
|publisher= [[Institute of Statistical Mathematics]]
|issn= 0020-3157
|doi= 10.1007/BF02480318
|url= http://www.ism.ac.jp/editsec/aism/pdf/032_1_0121.pdf
|format= PDF
|accessdate= February 18, 2013
|ref = harv
}}
*{{cite journal
| last1 = Parr
| first1 = William C.
| last2 = Schucany
| first2 = William R.
| year = 1980
| title = Minimum Distance and Robust Estimation
| journal = Journal of the American Statistical Association
| volume = 75
| issue = 371
| pages = 616–624
| publisher = [[American Statistical Association]]
| issn = 0162-1459
| doi = 10.1080/01621459.1980.10477522
| jstor = 2287658
| ref = harv
}}
*{{cite journal
|last= Wolfowitz
|first= J.
|authorlink=Jacob Wolfowitz
|date=March 1957
|title= The minimum distance method
|journal= The Annals of Mathematical Statistics
|volume= 28
|issue= 1
|pages= 75–88
|publisher= [[Institute of Mathematical Statistics]]
|issn= 0020-3157
|doi= 10.1214/aoms/1177707038
|url= http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aoms/1177707038
|format= PDF
|accessdate= February 18, 2013
}}
{{refend}}
 
{{Statistics|inference}}
 
{{DEFAULTSORT:Minimum distance estimation}}
<!-- Categories -->
[[Category:Estimation theory]]
[[Category:Statistical distance measures]]
[[Category:Mathematical modeling]]

Latest revision as of 23:12, 28 July 2014

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