Pearson distribution

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.

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.

This need became apparent when trying to fit known theoretical models to observed data that exhibited skewness.

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.

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).

(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.

The diagram shows which Pearson type a given concrete distribution (identified by a point (β1, β2)) belongs to.

[2] 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.

A Pearson density p is defined to be any valid solution to the differential equation (cf.

Pearson 1895, p. 381) with: According to Ord,[3] Pearson devised the underlying form of Equation (1) on the basis of, firstly, the formula for the derivative of the logarithm of the density function of the normal distribution (which gives a linear function) and, secondly, from a recurrence relation for values in the probability mass function of the hypergeometric distribution (which yields the linear-divided-by-quadratic structure).

Applying these substitutions, the quadratic function (2) is transformed into The absence of real roots is obvious from this formulation, because α2 is necessarily positive.

Then Finally, let Applying these substitutions, we obtain the parametric function: This unnormalized density has support on the entire real line.

One parameter was lost when we chose to find the solution to the differential equation (1) as a function of y rather than x.

An alternative parameterization (and slight specialization) of the type VII distribution is obtained by letting which requires m > 3/2.

This entails a minor loss of generality but ensures that the variance of the distribution exists and is equal to σ2.

If m approaches infinity as λ and σ are held constant, the normal distribution arises as a special case: This is the density of a normal distribution with mean λ and standard deviation σ.

It is convenient to require that m > 5/2 and to let This is another specialization, and it guarantees that the first four moments of the distribution exist.

More specifically, the Pearson type VII distribution parameterized in terms of (λ, σ, γ2) has a mean of λ, standard deviation of σ, skewness of zero, and positive excess kurtosis of γ2.

The Pearson type VII distribution is equivalent to the non-standardized Student's t-distribution with parameters ν > 0, μ, σ2 by applying the following substitutions to its original parameterization: Observe that the constraint m > 1/2 is satisfied.

In particular, the standard Student's t-distribution arises as a subcase, when μ = 0 and σ2 = 1, equivalent to the following substitutions: The density of this restricted one-parameter family is a standard Student's t: If the quadratic function (2) has a non-negative discriminant (

Using the substitution we obtain the following solution to the differential equation (1): Since this density is only known up to a hidden constant of proportionality, that constant can be changed and the density written as follows: The Pearson type I distribution (a generalization of the beta distribution) arises when the roots of the quadratic equation (2) are of opposite sign, that is,

, which yields a solution in terms of y that is supported on the interval (0, 1): One may define: Regrouping constants and parameters, this simplifies to: Thus

The Pearson type II distribution is used in computing the table of significant correlation coefficients for Spearman's rank correlation coefficient when the number of items in a series is less than 100 (or 30, depending on some sources).

QPDs and metalogs can provide greater shape and bounds flexibility than the Pearson system.

Examples of modern alternatives to the Pearson skewness-vs-kurtosis diagram are: (i) https://github.com/SchildCode/PearsonPlot and (ii) the "Cullen and Frey graph" in the statistical application R. These models are used in financial markets, given their ability to be parametrized in a way that has intuitive meaning for market traders.

A number of models are in current use that capture the stochastic nature of the volatility of rates, stocks, etc.,[which?

In the United States, the Log-Pearson III is the default distribution for flood frequency analysis.

[5] Recently, there have been alternatives developed to the Pearson distributions that are more flexible and easier to fit to data.