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Pearson Distribution / Pearson Curves: Definition, Examples

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What is the Pearson Distribution?

The Pearson distribution (sometimes called the Pearson system of distributions) is a family of unimodal continuous probability distribution functions that satisfy the following differential equation:
pearson distribution

Pearson described twelve families of distributions as solutions to the equation. His original paper, (1895, p. 360) identified four (numbered I through IV) as well as type V — now known as the inverse gamma distribution: types VI through XII were identified in later papers. The family now includes the normal distribution.

PDF

Pearson curves are graphs of p(x) as a function of f. The general Pearson family of curves can show cases of the gamma distribution, log-normal distribution and inverse gamma distribution (Lahcene, 2013) Special cases of the different types give rise to various known distributions. For example:

As such, it is one of the more challenging distributions to find parameters for, finding the pdf and fitting is usually performed with software (such as the PearsonDS package in R) using maximum likelihood and method of moments. The density and range for the twelve distributions are as follows (from Ord, 2006):

Type Density Support
I (1 + x)m1(1 – x)m2 (-1 ≤ x ≤ 1)
II (1 – x2)m (-1 ≤ x ≤ 1)
III xmexp(-x) (0 ≤ x < ∞)
IV (1 + x2)-mx exp(-υ tan-1(x)) (-∞ < x < ∞)
V x-mexp(-x-1) (0 ≤ x < ∞)
VI xm2(1 + x)-m1 (0 ≤ x < ∞)
VII (1 + x2)-m (-∞ < x < ∞)
VIII (1 + x)-m (0 ≤ x ≤ 1)
IX (1 + x)m (0 ≤ x ≤ 1)
X e-x (0 ≤ x < ∞)
XI x-m (1 ≤ x < ∞)
XII [(g + x)/ (g – x)]h (-g ≤ x ≤ g)
normal exp(-½x2) (-∞ < x < ∞)

References:
Lahcene, B. On Pearson families of distributions and its applications. African Journal of Mathematics and Computer Science Research. Vol. 6(5), pp. 108-117, May 2013. Retrieved July 30, 2017 from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.402.3236&rep=rep1&type=pdf

Ord, J.K. (2006). Oakes’s Test of Concordance to Preference Functions. Retrieved 7/31/2017 from: http://onlinelibrary.wiley.com/doi/10.1002/0471667196.ess1939.pub2/abstract

Pearson K (1895). “Contributions to the mathematical theory of evolution, II: Skew variation in homogeneous material.” Philos. Trans. Royal Soc. London, ARRAY 186:343-414

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