Editing Maximum likelihood estimation (section)

== History ==
[[File:Youngronaldfisher2.JPG|thumb|right|[[Ronald Fisher]] in 1913]]
Early users of maximum likelihood include [[Carl Friedrich Gauss]], [[Pierre-Simon Laplace]], [[Thorvald N. Thiele]], and [[Francis Ysidro Edgeworth]].<ref>
{{cite journal
  | doi = 10.2307/2339293
  | last = Edgeworth | first = Francis Y. | author-link = Francis Ysidro Edgeworth
  | title = On the probable errors of frequency-constants
  |date=Sep 1908
  | ref= CITEREFEdgeworthSeptember_1908
  | journal = Journal of the Royal Statistical Society
  | volume = 71 | issue = 3
  | pages = 499–512
  | jstor = 2339293
  | url = https://zenodo.org/record/1449468 }}</ref><ref>{{cite journal
  | doi = 10.2307/2339378
  | last = Edgeworth | first = Francis Y.
  | title = On the probable errors of frequency-constants
  |date=Dec 1908
  | ref= CITEREFEdgeworthDecember_1908
  | journal = Journal of the Royal Statistical Society
  | volume = 71 | issue = 4
  | pages = 651–678
  | jstor = 2339378
  | url = https://zenodo.org/record/1449468
}}
</ref> It was [[Ronald Fisher]] however, between 1912 and 1922, who singlehandedly created the modern version of the method.<ref name="Pfanzagl">
{{cite book
 |last1=Pfanzagl |first1=Johann
 |year=1994
 |title=Parametric Statistical Theory
 |publisher=[[Walter de Gruyter]]
 |isbn=978-3-11-013863-4
 |pages=207–208
 |doi=10.1515/9783110889765
 |mr=1291393
}}</ref><ref>{{Cite journal |last=Hald |first=Anders |date=1999 |title=On the History of Maximum Likelihood in Relation to Inverse Probability and Least Squares |url=https://www.jstor.org/stable/2676741 |journal=Statistical Science |volume=14 |issue=2 |pages=214–222 |issn=0883-4237}}</ref>

Maximum-likelihood estimation finally transcended [[heuristic]] justification in a proof published by [[Samuel S. Wilks]] in 1938, now called [[Wilks' theorem]].<ref>{{cite journal |last=Wilks |first=S.S. |year=1938 |title=The large-sample distribution of the likelihood ratio for testing composite hypotheses |journal=[[Annals of Mathematical Statistics]] |volume=9 |pages=60–62 |doi=10.1214/aoms/1177732360 |doi-access=free}}</ref> The theorem shows that the error in the logarithm of likelihood values for estimates from multiple independent observations is asymptotically [[chi-squared distribution|''χ''<sup>&thinsp;2</sup>-distributed]], which enables convenient determination of a [[confidence region]] around any estimate of the parameters. The only difficult part of Wilks' proof depends on the expected value of the [[Fisher information]] matrix, which is provided by a theorem proven by Fisher.<ref>
{{cite book
 |last=Owen |first=Art B.
 |year=2001
 |title=Empirical Likelihood
 |place=London, UK; Boca Raton, FL
 |publisher=Chapman & Hall; CRC Press
 |isbn=978-1-58488-071-4
}}
</ref> Wilks continued to improve on the generality of the theorem throughout his life, with his most general proof published in 1962.<ref>
{{cite book
 |last=Wilks |first=Samuel S.
 |year=1962
 |title=Mathematical Statistics
 |place=New York, NY
 |publisher=John Wiley & Sons
 |isbn=978-0-471-94650-2
}}
</ref>

Reviews of the development of maximum likelihood estimation have been provided by a number of authors.<ref>
{{cite journal
  | doi = 10.1214/aos/1176343456
  | last = Savage | first = Leonard J. | author-link = Leonard J. Savage
  | title = On rereading R.A. Fisher
  | year = 1976
  | journal = The Annals of Statistics
  | volume = 4 | issue = 3
  | pages = 441–500
  | jstor = 2958221
  | doi-access = free
}}
</ref><ref>
{{cite journal
  | doi = 10.1214/aos/1176343457
  | last = Pratt | first = John W.
  | title = F. Y. Edgeworth and R. A. Fisher on the efficiency of maximum likelihood estimation
  | year = 1976
  | journal = The Annals of Statistics
  | volume = 4 | issue = 3
  | pages = 501–514
  | jstor = 2958222
  | doi-access = free
}}
</ref><ref>
{{cite journal
  | doi = 10.2307/2344804
  | last = Stigler | first = Stephen M. | author-link = Stephen M. Stigler
  | title = Francis Ysidro Edgeworth, statistician
  | year = 1978
  | journal = Journal of the Royal Statistical Society, Series A
  | volume = 141 | issue = 3
  | pages = 287–322
  | jstor = 2344804
  }}
</ref><ref>
{{cite book
  | last = Stigler  | first = Stephen M.
  | title = The history of statistics: the measurement of uncertainty before 1900
  | year = 1986
  | publisher = Harvard University Press
  | isbn = 978-0-674-40340-6
  | url = https://archive.org/details/historyofstatist00stig
}}
</ref><ref>
{{cite book
  | last = Stigler | first = Stephen M.
  | title = Statistics on the table: the history of statistical concepts and methods
  | year = 1999
  | publisher = Harvard University Press
  | isbn = 978-0-674-83601-3
  }}</ref><ref>{{cite book
  | last = Hald | first = Anders | author-link=Anders Hald
  | title = A history of mathematical statistics from 1750 to 1930
  | year = 1998
  | publisher = Wiley
  | location = New York, NY
  | isbn = 978-0-471-17912-2
  }}
</ref><ref>
{{cite journal
  | last = Hald | first = Anders
  | title = On the history of maximum likelihood in relation to inverse probability and least squares
  | year = 1999
  | journal = Statistical Science
  | volume = 14 | issue = 2
  | pages =214–222
  | jstor = 2676741
  | doi=10.1214/ss/1009212248
  | url = http://projecteuclid.org/download/pdf_1/euclid.ss/1009212248
  | doi-access = free
}}
</ref><ref>
{{cite journal
  | last = Aldrich | first = John
  | title = R.A. Fisher and the making of maximum likelihood 1912–1922
  | year = 1997
  | journal = Statistical Science
  | volume = 12 | issue = 3
  | pages = 162–176
  | doi = 10.1214/ss/1030037906
  | mr = 1617519
  | doi-access = free
}}
</ref>