Editing Beta distribution (section)

==History==
[[Thomas Bayes]], in a posthumous paper <ref name="ThomasBayes">{{cite journal|last=Bayes|first=Thomas|author2=communicated by Richard Price|title=An Essay towards solving a Problem in the Doctrine of Chances|journal=Philosophical Transactions of the Royal Society|year=1763|volume=53|pages=370–418|jstor=105741|doi=10.1098/rstl.1763.0053|doi-access=free}}</ref> published in 1763 by [[Richard Price]], obtained a beta distribution as the density of the probability of success in Bernoulli trials (see {{section link||Applications, Bayesian inference}}), but the paper does not analyze any of the moments of the beta distribution or discuss any of its properties.

[[File:Karl Pearson 2.jpg|thumb|220px|[[Karl Pearson]] analyzed the beta distribution as the solution Type I of Pearson distributions ]]

The first systematic modern discussion of the beta distribution is probably due to [[Karl Pearson]].<ref>{{Cite journal | last1 = Yule | first1 = G. U. | author-link1 = Udny Yule| last2 = Filon | first2 = L. N. G. | doi = 10.1098/rsbm.1936.0007 | title = Karl Pearson. 1857–1936 | journal = [[Obituary Notices of Fellows of the Royal Society]] | volume = 2 | issue = 5 | pages = 72 | year = 1936 | jstor = 769130| title-link = Karl Pearson }}</ref><ref name=rscat>{{cite web|url=http://www2.royalsociety.org/DServe/dserve.exe?dsqIni=Dserve.ini&dsqApp=Archive&dsqCmd=Show.tcl&dsqDb=Persons&dsqPos=0&dsqSearch=%28%28text%29%3D%27%20%20Pearson%3A%20Karl%20%281857%20-%201936%29%20%20%27%29%29|access-date=2011-07-01|title=Library and Archive catalogue|work=Sackler Digital Archive|publisher=Royal Society|archive-url=https://web.archive.org/web/20111025030931/http://www2.royalsociety.org/DServe/dserve.exe?dsqIni=Dserve.ini&dsqApp=Archive&dsqCmd=Show.tcl&dsqDb=Persons&dsqPos=0&dsqSearch=%28%28text%29%3D%27%20%20Pearson%3A%20Karl%20%281857%20-%201936%29%20%20%27%29)|archive-date=2011-10-25|url-status=dead}}</ref> In Pearson's papers<ref name=Pearson/><ref name=Pearson1895/> the beta distribution is couched as a solution of a differential equation: [[Pearson distribution|Pearson's Type I distribution]] which it is essentially identical to except for arbitrary shifting and re-scaling (the beta and Pearson Type I distributions can always be equalized by proper choice of parameters). In fact, in several English books and journal articles in the few decades prior to World War II, it was common to refer to the beta distribution as Pearson's Type I distribution.  [[William Palin Elderton|William P. Elderton]] in his 1906 monograph "Frequency curves and correlation"<ref name=Elderton1906>{{cite book|last=Elderton|first=William Palin|title=Frequency-Curves and Correlation |year=1906|publisher=Charles and Edwin Layton (London)|url=https://archive.org/details/frequencycurvesc00elderich}}</ref> further analyzes the beta distribution as Pearson's Type I distribution, including a full discussion of the method of moments for the four parameter case, and diagrams of (what Elderton describes as) U-shaped, J-shaped, twisted J-shaped, "cocked-hat" shapes, horizontal and angled straight-line cases.  Elderton wrote "I am chiefly indebted to Professor Pearson, but the indebtedness is of a kind for which it is impossible to offer formal thanks."  [[William Palin Elderton|Elderton]] in his 1906 monograph <ref name=Elderton1906/> provides an impressive amount of information on the beta distribution, including equations for the origin of the distribution chosen to be the mode, as well as for other Pearson distributions: types I through VII. Elderton also included a number of appendixes, including one appendix ("II") on the beta and gamma functions. In later editions, Elderton added equations for the origin of the distribution chosen to be the mean, and analysis of Pearson distributions VIII through XII.

As remarked by Bowman and Shenton<ref name="BowmanShenton"/> "Fisher and Pearson had a difference of opinion in the approach to (parameter) estimation, in particular relating to (Pearson's method of) moments and (Fisher's method of) maximum likelihood in the case of the Beta distribution." Also according to Bowman and Shenton, "the case of a Type I (beta distribution) model being the center of the controversy was pure serendipity. A more difficult model of 4 parameters would have been hard to find." The long running public conflict of Fisher with Karl Pearson can be followed in a number of articles in prestigious journals.  For example, concerning the estimation of the four parameters for the beta distribution, and Fisher's criticism of Pearson's method of moments as being arbitrary, see Pearson's article "Method of moments and method of maximum likelihood" <ref name=Pearson1936>{{cite journal|last=Pearson|first=Karl|title=Method of moments and method of maximum likelihood|journal=Biometrika|date=June 1936|volume=28|issue=1/2|doi=10.2307/2334123|pages=34–59|jstor=2334123}}</ref> (published three years after his retirement from University College, London, where his position had been divided between Fisher and Pearson's son Egon) in which Pearson writes "I read (Koshai's paper in the Journal of the Royal Statistical Society, 1933) which as far as I am aware is the only case at present published of the application of Professor Fisher's method. To my astonishment that method depends on first working out the constants of the frequency curve by the (Pearson) Method of Moments and then superposing on it, by what Fisher terms "the Method of Maximum Likelihood" a further approximation to obtain, what he holds, he will thus get, 'more efficient values' of the curve constants".

David and Edwards's treatise on the history of statistics<ref name="David History">{{cite book|last=David|first=H. A. and A.W.F. Edwards|title=Annotated Readings in the History of Statistics|year=2001|publisher=Springer; 1 edition|isbn=978-0387988443}}</ref> cites the first modern treatment of the beta distribution, in 1911,<ref>{{cite journal |last=Gini |first=Corrado |title=Considerazioni Sulle Probabilità Posteriori e Applicazioni al Rapporto dei Sessi Nelle Nascite Umane |journal=Studi Economico-Giuridici della Università de Cagliari |year=1911 |volume=Anno III |issue=reproduced in Metron 15, 133,171, 1949 |pages=5–41}}</ref>  using the beta designation that has become standard, due to [[Corrado Gini]], an Italian [[statistician]], [[demography|demographer]], and [[sociology|sociologist]], who developed the [[Gini coefficient]]. [[Norman Lloyd Johnson|N.L.Johnson]] and [[Samuel Kotz|S.Kotz]], in their comprehensive and very informative monograph<ref>{{cite book|editor-last=Johnson|editor-first=Norman L. and Samuel Kotz|title=Leading Personalities in Statistical Sciences: From the Seventeenth Century to the Present (Wiley Series in Probability and Statistics|year=1997|publisher=Wiley|isbn=978-0471163817}}</ref>  on leading historical personalities in statistical sciences credit [[Corrado Gini]]<ref>{{cite web|last=Metron journal. |title=Biography of Corrado Gini |url=http://www.metronjournal.it/storia/ginibio.htm |publisher=Metron Journal |access-date=2012-08-18 |url-status=dead|archive-url=https://web.archive.org/web/20120716202225/http://www.metronjournal.it/storia/ginibio.htm |archive-date=2012-07-16 }}</ref>  as "an early Bayesian...who dealt with the problem of eliciting the parameters of an initial Beta distribution, by singling out techniques which anticipated the advent of the so-called empirical Bayes approach."