The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History The Pearson system was originally devised in an effort to model visibly skewed observations. It was well known at the time how to adjust a theoretical model to fit the first two cumulants or moments of observed data: any probability distribution can be extended straightforwardly to form a location-scale family. Except in pathological cases, a location-scale family can be made to fit the observed mean (first cumulant) and variance (second cumulant) arbitrarily well. However, it was not known how to construct probability distributions in which the skewness (standardized third cumulant) and kurtosis (standardized fourth cumulant) could be adjusted equally freely. This need became apparent when trying to fit known theoretical models to observed data that exhibited skewness. Pearson's examples include survival data, which are usually asymmetric. In his original paper, Pearson (1895, p. 360) identified four types of distributions (numbered I through IV) in addition to the normal distribution (which was originally known as type V). The classification depended on whether the distributions were supported on a bounded interval, on a half-line, or on the whole real line; and whether they were potentially skewed or necessarily symmetric. A second paper (Pearson 1901) fixed two omissions: it redefined the type V distribution (originally just the normal distribution, but now the inverse-gamma distribution) and introduced the type VI distribution. Together the first two papers cover the five main types of the Pearson system (I, III, IV, V, and VI). In a third paper, Pearson (1916) introduced further special cases and subtypes (VII through XII). Rhind (1909, pp. 430–432) devised a simple way of visualizing the parameter space of the Pearson system, which was subsequently adopted by Pearson (1916, plate 1 and pp. 430ff., 448ff.). The Pearson types are characterized by two quantities, commonly referred to as β1 and β2. The first is the square of the skewness: β1 = γ12 where γ1 is the skewness, or third standardized moment. The second is the traditional kurtosis, or fourth standardized moment: β2 = γ2 + 3. (Modern treatments define kurtosis γ2 in terms of cumulants instead of moments, so that for a normal distribution we have γ2 = 0 and β2 = 3. Here we follow the historical precedent and use β2.) The diagram shows which Pearson type a given concrete distribution (identified by a point (β1, β2)) belongs to. Many of the skewed or non-mesokurtic distributions familiar to statisticians today were still unknown in the early 1890s. What is now known as the beta distribution had been used by Thomas Bayes as a posterior distribution of the parameter of a Bernoulli distribution in his 1763 work on inverse probability. The beta distribution gained prominence due to its membership in Pearson's system and was known until the 1940s as the Pearson type I distribution. (Pearson's type II distribution is a special case of type I, but is usually no longer singled out.) The gamma distribution originated from Pearson's work (Pearson 1893, p. 331; Pearson 1895, pp. 357, 360, 373–376) and was known as the Pearson type III distribution, before acquiring its modern name in the 1930s and 1940s. Pearson's 1895 paper introduced the type IV distribution, which contains Student's t-distribution as a special case, predating William Sealy Gosset's subsequent use by several years. His 1901 paper introduced the inverse-gamma distribution (type V) and the beta prime distribution (type VI).
Definition A Pearson density p is defined to be any valid solution to the differential equation (cf. Pearson 1902, p. 277)
p ′ ( x ) p ( x ) + A ( x ) B ( x ) = 0 ( 1 ) {\displaystyle {\frac {p'(x)}{p(x)}}+{\frac {A(x)}{B(x)}}=0\qquad (1)}
where:
A(x) = a1 + a2x is a polynomial of degree at most 1, B(x) = b0 + b1x + b2x2 with B ≠ 0 is a non-zero polynomial of degree at most 2, (a2, b2) ≠ (0, 0): if (a2, b2) = (0, 0) but (a1, b1) ≠ (0, 0), multiply A and B by x+u with arbitrary finite u ≠ b0⁄b1, if (a2, b2) = (0, 0) and (a1, b1) = (0, 0) (hence A = 0), multiply A and B by (x+u)(x+v) with arbitrary finite u and v satisfying u ≠ v, the domain I of p is an open interval (xmin, xmax) bounded by infinities and/or real zeros of B but not containing any of them, i.e.: if B has no real zeros, then I = R {\displaystyle I=\mathbb {R} } , if B has one real zero x0, then I = (−∞, x0) or I = (x0, +∞), if B has two real zeros x− < x+, then I = (−∞, x−) or I = (x−, x+) or I = (x+, +∞). It's why, for instance, the half-normal and truncated normal distributions are not Pearson distributions (their domain doesn't match the above criteria) even though they satisfy equation (1). Dimensional analysis shows that coefficients ak and bk have dimension [AX−k], with [A] an arbitrary dimension and [X] the dimension of x, hence the shift in the indices of ak. If the distribution has moments up to order at least 4, with mean μ and standard deviation σ, we have (cf. Pearson 1916, p. 437, and Carver 1924, pp. 103–104)
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