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, VI, V, and IV). 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 and . The first is the square of the skewness: where is the skewness, or third standardized moment. The second is the traditional kurtosis, or fourth standardized moment: . (Modern treatments define kurtosis in terms of cumulants instead of moments, so that for a normal distribution we have and . Here we follow the historical precedent and use .) The diagram on the right shows which Pearson type a given concrete distribution (identified by a point ) belongs to.
Many of the skewed and/or non-mesokurtic distributions familiar to us 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).
Read more about this topic: Pearson Distribution
Famous quotes containing the word history:
“Yet poetry, though the last and finest result, is a natural fruit. As naturally as the oak bears an acorn, and the vine a gourd, man bears a poem, either spoken or done. It is the chief and most memorable success, for history is but a prose narrative of poetic deeds.”
—Henry David Thoreau (1817–1862)
“The history of work has been, in part, the history of the worker’s body. Production depended on what the body could accomplish with strength and skill. Techniques that improve output have been driven by a general desire to decrease the pain of labor as well as by employers’ intentions to escape dependency upon that knowledge which only the sentient laboring body could provide.”
—Shoshana Zuboff (b. 1951)
“What you don’t understand is that it is possible to be an atheist, it is possible not to know if God exists or why He should, and yet to believe that man does not live in a state of nature but in history, and that history as we know it now began with Christ, it was founded by Him on the Gospels.”
—Boris Pasternak (1890–1960)