Beta Distribution - History

History

The first systematic, modern discussion of the beta distribution is probably due to Karl Pearson FRS (27 March 1857 – 27 April 1936), an influential English mathematician who has been credited with establishing the discipline of mathematical statistics. In Pearson's papers the beta distribution is couched as a solution of a differential equation: Pearson's Type I distribution. The beta distribution is essentially identical to Pearson's Type I distribution 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 P. Elderton (1877–1962) in his 1906 monograph "Frequency curves and correlation" 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." Elderton in his 1906 monograph 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 "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." Ronald Fisher (17 February 1890 – 29 July 1962) was one of the giants of statistics in the first half of the 20th century, and his long running public conflict 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" (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 cites the first modern treatment of the beta distribution, in 1911, using the beta designation that has become standard, due to Corrado Gini,(May 23, 1884 – March 13, 1965), an Italian statistician, demographer, and sociologist, who developed the Gini coefficient. N.L.Johnson and S.Kotz, in their comprehensive and very informative monograph on leading historical personalities in statistical sciences credit Corrado Gini 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." Bayes, in a posthumous paper published in 1763, obtained a beta distribution as the density of the probability of success in Bernoulli trials (see the section titled "Applications, Bayesian inference" in this article), but the paper does not analyze any of the moments of the beta distribution or discuss any of its properties.