Pearson distribution

File:Pearson system.png
Diagram of the Pearson system, showing distributions of types I, III, VI, V, and IV in terms of ${\displaystyle \beta _{1}}$ (squared skewness) and ${\displaystyle \beta _{2}}$ (traditional kurtosis)

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, 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 ${\displaystyle \beta _{1}}$ and ${\displaystyle \beta _{2}}$. The first is the square of the skewness: ${\displaystyle \beta _{1}=\gamma _{1}^{2}}$ where ${\displaystyle \gamma _{1}}$ is the skewness, or third standardized moment. The second is the traditional kurtosis, or fourth standardized moment: ${\displaystyle \beta _{2}=\gamma _{2}+3}$. (Modern treatments define kurtosis ${\displaystyle \gamma _{2}}$ in terms of cumulants instead of moments, so that for a normal distribution we have ${\displaystyle \gamma _{2}=0}$ and ${\displaystyle \beta _{2}=3}$. Here we follow the historical precedent and use ${\displaystyle \beta _{2}}$.) The diagram on the right shows which Pearson type a given concrete distribution (identified by a point ${\displaystyle (\beta _{1},\beta _{2})}$) 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. [1] (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. [2] Pearson's 1895 paper introduced the type IV distribution, which contains Student's t-distribution as a special case, predating William 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 1895, p. 381)

${\displaystyle {\frac {p'(x)}{p(x)}}+{\frac {a+x-\lambda }{b_{2}(x-\lambda )^{2}+b_{1}(x-\lambda )+b_{0}}}=0.\qquad (1)\!}$

with : ${\displaystyle b_{0}={\frac {4\beta _{2}-3\beta _{1}}{10\beta _{2}-12\beta _{1}-18}}\mu _{2}}$

${\displaystyle a=b_{1}={\sqrt {\mu _{2}\beta _{1}}}{\frac {\beta _{2}+3}{10\beta 2-12\beta _{1}-18}}}$

${\displaystyle b_{2}={\frac {2\beta _{2}-3\beta _{1}-6}{10\beta 2-12\beta _{1}-18}}}$

The parameter a0 determines a stationary point, and hence under some conditions a mode of the distribution, since

${\displaystyle p'(a_{0})=0\!}$

follows directly from the differential equation.

Since we are confronted with a linear differential equation with variable coefficients, its solution is straightforward:

${\displaystyle p(x)\propto \exp \left(-\!\int \!\!{\frac {x-a}{b_{2}x^{2}+b_{1}x+b_{0}}}\,\mathrm {d} x\right).\!}$

The integral in this solution simplifies considerably when certain special cases of the integrand are considered. Pearson (1895, p. 367) distinguished two main cases, determined by the sign of the discriminant (and hence the number of real roots) of the quadratic function

${\displaystyle f(x)=b_{2}\,x^{2}+b_{1}\,x+b_{0}.\qquad (2)\!}$

Case 1, negative discriminant: The Pearson type IV distribution

If the discriminant of the quadratic function (2) is negative (${\displaystyle b_{1}^{2}-4b_{2}b_{0}<0}$), it has no real roots. Then define

${\displaystyle y=x+{\frac {b_{1}}{2\,b_{2}}}\!}$  and
${\displaystyle \alpha ={\frac {\sqrt {4\,b_{2}\,b_{0}-b_{1}^{2}\,}}{2\,b_{2}}}.\!}$

Observe that ${\displaystyle \alpha }$ is a well-defined real number and ${\displaystyle \alpha \neq 0}$, because by assumption ${\displaystyle 4b_{2}b_{0}-b_{1}^{2}>0}$ and therefore ${\displaystyle b_{2}\neq 0}$. Applying these substitutions, the quadratic function (2) is transformed into

${\displaystyle f(x)=b_{2}\,(y^{2}+\alpha ^{2}).\!}$

The absence of real roots is obvious from this formulation, because ${\displaystyle \alpha ^{2}}$ is necessarily positive.

We now express the solution to the differential equation (1) as a function of y:

${\displaystyle p(y)\propto \exp \left(-{\frac {1}{b_{2}}}\,\int {\frac {y-{\frac {b_{1}}{2\,b_{2}}}-a}{y^{2}+\alpha ^{2}}}\,\mathrm {d} y\right).\!}$

Pearson (1895, p. 362) called this the "trigonometrical case", because the integral

${\displaystyle \int {\frac {y-{\frac {2\,b_{2}\,a+b_{1}}{2\,b_{2}}}}{y^{2}+\alpha ^{2}}}\,\mathrm {d} y={\frac {1}{2}}\ln(y^{2}+\alpha ^{2})-{\frac {2\,b_{2}\,a+b_{1}}{2\,b_{2}\,\alpha }}\arctan \left({\frac {y}{\alpha }}\right)+C_{0}\!}$

involves the inverse trigonometic arctan function. Then

${\displaystyle p(y)\propto \exp \left[-{\frac {1}{2\,b_{2}}}\ln \!\left(1+{\frac {y^{2}}{\alpha ^{2}}}\right)-{\frac {\ln \alpha }{2\,b_{2}}}+{\frac {2\,b_{2}\,a+b_{1}}{2\,b_{2}^{2}\,\alpha }}\arctan \left({\frac {y}{\alpha }}\right)+C_{1}\right]\!}$

Finally, let

${\displaystyle m={\frac {1}{2\,b_{2}}}\!}$  and
${\displaystyle \nu =-{\frac {2\,b_{2}\,a+b_{1}}{2\,b_{2}^{2}\,\alpha }}\!}$

Applying these substitutions, we obtain the parametric function:

${\displaystyle p(y)\propto \left[1+{\frac {y^{2}}{\alpha ^{2}}}\right]^{-m}\exp \left[-\nu \arctan \left({\frac {y}{\alpha }}\right)\right]\!}$

This unnormalized density has support on the entire real line. It depends on a scale parameter ${\displaystyle \alpha >0}$ and shape parameters ${\displaystyle m>1/2}$ and ${\displaystyle \nu }$. One parameter was lost when we chose to find the solution to the differential equation (1) as a function of y rather than x. We therefore reintroduce a fourth parameter, namely the location parameter λ. We have thus derived the density of the Pearson type IV distribution:

${\displaystyle p(x)={\frac {\left|{\frac {\Gamma \!\left(m+{\frac {\nu }{2}}i\right)}{\Gamma (m)}}\right|^{2}}{\alpha \,\mathrm {\mathrm {B} } \!\left(m-{\frac {1}{2}},{\frac {1}{2}}\right)}}\left[1+\left({\frac {x-\lambda }{\alpha }}\right)^{\!2\,}\right]^{-m}\exp \left[-\nu \arctan \left({\frac {x-\lambda }{\alpha }}\right)\right].\!}$

The normalizing constant involves the complex Gamma function (Γ) and the Beta function (B).

The Pearson type VII distribution

File:Pearson type VII distribution PDF.png
Plot of Pearson type VII densities with ${\displaystyle \lambda =0}$, ${\displaystyle \sigma =1}$, and: ${\displaystyle \gamma _{2}=\infty }$ (red); ${\displaystyle \gamma _{2}=4}$ (blue); and ${\displaystyle \gamma _{2}=0}$ (black)

The shape parameter ν of the Pearson type IV distribution controls its skewness. If we fix its value at zero, we obtain a symmetric three-parameter family. This special case is known as the Pearson type VII distribution (cf. Pearson 1916, p. 450). Its density is

${\displaystyle p(x)={\frac {1}{\alpha \,\mathrm {\mathrm {B} } \!\left(m-{\frac {1}{2}},{\frac {1}{2}}\right)}}\left[1+\left({\frac {x-\lambda }{\alpha }}\right)^{\!2\,}\right]^{-m},\!}$

where B is the Beta function.

An alternative parameterization (and slight specialization) of the type VII distribution is obtained by letting

${\displaystyle \alpha =\sigma \,{\sqrt {2\,m-3}},\!}$

which requires ${\displaystyle m>3/2}$. This entails a minor loss of generality but ensures that the variance of the distribution exists and is equal to ${\displaystyle \sigma ^{2}}$. Now the parameter m only controls the kurtosis of the distribution. If m approaches infinity as λ and σ are held constant, the normal distribution arises as a special case:

${\displaystyle \lim _{m\to \infty }{\frac {1}{\sigma \,{\sqrt {2\,m-3}}\,\mathrm {\mathrm {B} } \!\left(m-{\frac {1}{2}},{\frac {1}{2}}\right)}}\left[1+\left({\frac {x-\lambda }{\sigma \,{\sqrt {2\,m-3}}}}\right)^{\!2\,}\right]^{-m}\!}$
${\displaystyle ={\frac {1}{\sigma \,{\sqrt {2}}\,\Gamma \!\left({\frac {1}{2}}\right)}}\times \lim _{m\to \infty }{\frac {\Gamma (m)}{\Gamma \!\left(m-{\frac {1}{2}}\right){\sqrt {m-{\frac {3}{2}}}}}}\times \lim _{m\to \infty }\left[1+{\frac {\left({\frac {x-\lambda }{\sigma }}\right)^{2}}{2\,m-3}}\right]^{-m}\!}$
${\displaystyle ={\frac {1}{\sigma {\sqrt {2\,\pi }}}}\times 1\times \exp \!\left[-{\frac {1}{2}}\left({\frac {x-\lambda }{\sigma }}\right)^{\!2\,}\right]\!}$

This is the density of a normal distribution with mean λ and standard deviation σ.

It is convenient to require that ${\displaystyle m>5/2}$ and to let

${\displaystyle m={\frac {5}{2}}+{\frac {3}{\gamma _{2}}}.\!}$

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 ${\displaystyle (\lambda ,\sigma ,\gamma _{2})}$ has a mean of λ, standard deviation of σ, skewness of zero, and excess kurtosis of ${\displaystyle \gamma _{2}}$.

Student's t-distribution

The Pearson type VII distribution subsumes Student's t-distribution, and hence also the Cauchy distribution. Student's t-distribution arises as the result of applying the following substitutions to its original parameterization:

${\displaystyle \lambda =0,\!}$
${\displaystyle \alpha ={\sqrt {\nu }},\!}$  and
${\displaystyle m={\frac {\nu +1}{2}},\!}$

where ${\displaystyle \nu >0}$. Observe that the constraint ${\displaystyle m>1/2}$ is satisfied. The density of this restricted one-parameter family is

${\displaystyle p(x)={\frac {1}{{\sqrt {\nu }}\,\mathrm {\mathrm {B} } \!\left({\frac {\nu }{2}},{\frac {1}{2}}\right)}}\left[1+{\frac {x^{2}}{\nu }}\right]^{-{\frac {\nu +1}{2}}},\!}$

which is easily recognized as the density of Student's t-distribution.

Case 2, non-negative discriminant

If the quadratic function (2) has a non-negative discriminant (${\displaystyle b_{1}^{2}-4b_{2}b_{0}\geq 0}$), it has real roots a1 and a2 (not necessarily distinct):

${\displaystyle a_{1}={\frac {-b_{1}-{\sqrt {b_{1}^{2}-4b_{2}b_{0}}}}{2b_{2}}},\!}$
${\displaystyle a_{2}={\frac {-b_{1}+{\sqrt {b_{1}^{2}-4b_{2}b_{0}}}}{2b_{2}}},\!}$

One have to define :

${\displaystyle m_{1}={\frac {a+a_{1}}{c_{2}(a_{2}-a_{1})}}\!}$
${\displaystyle m_{2}=-{\frac {a+a_{2}}{c_{2}(a_{2}-a_{1})}}\!}$
${\displaystyle C_{1}={\frac {c_{1}}{2c_{2}}}\!}$

In the presence of real roots the quadratic function (2) can be written as

${\displaystyle f(x)=b_{2}\,(x-a_{1})(x-a_{2}),\!}$

and the solution to the differential equation is therefore

${\displaystyle p(x)\propto \exp \left(-{\frac {1}{b_{2}}}\int \!\!{\frac {x-a}{(x-a_{1})(x-a_{2})}}\,\mathrm {d} x\right).\!}$

Pearson (1895, p. 362) called this the "logarithmic case", because the integral

${\displaystyle \int \!\!{\frac {x-a}{(x-a_{1})(x-a_{2})}}\,\mathrm {d} x={\frac {(a_{1}-a)\ln(x-a_{1})-(a_{2}-a)\ln(x-a_{2})}{a_{1}-a_{2}}}+C\!}$

involves only the logarithm function, and not the arctan function as in the previous case.

Using the substitution

${\displaystyle \nu ={\frac {1}{b_{2}\,(a_{1}-a_{2})}}\!}$

we obtain the following solution to the differential equation (1):

${\displaystyle p(x)\propto (x-r_{1})^{-\nu (a_{1}-a)}(x-a_{2})^{\nu (a_{2}-a)}.\!}$

Since this density is only known up to a hidden constant of proportionality, that constant can be changed and the density written as follows:

${\displaystyle p(x)\propto \left(1-{\frac {x}{a_{1}}}\right)^{-\nu (a_{1}-a)}\left(1-{\frac {x}{a_{2}}}\right)^{\nu (a_{2}-a)}\!}$

The Pearson type I and type II distribution

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, ${\displaystyle r_{1}<0. Then the solution p is supported on the interval ${\displaystyle (r_{1},r_{2})}$. Apply the substition

${\displaystyle x=a_{1}+y(a_{2}-a_{1})\qquad {\mbox{where}}\ 0

which yields a solution in terms of y that is supported on the interval ${\displaystyle (0,1)}$:

${\displaystyle p(y)\propto \left({\frac {a_{1}-a_{2}}{a_{1}}}\;y\right)^{(-a_{1}+a)\nu }\left({\frac {a_{2}-a_{1}}{a_{2}}}\;(1-y)\right)^{(a_{2}-a)\nu }.\!}$

Regrouping constants and parameters, this simplifies to:

${\displaystyle p(y)\propto y^{m_{1}}(1-y)^{m_{2}},\!}$

Thus ${\displaystyle {\frac {x-\lambda -a_{1}}{a_{2}-a_{1}}}\!}$ follows a ${\displaystyle beta(m_{1}+1,m_{2}+1)\!}$ with ${\displaystyle \lambda =\mu _{1}-(a_{2}-a_{1}){\frac {m_{1}+1}{m_{1}+m_{2}+2}}-a_{1}\!}$

It turns out that ${\displaystyle m_{1}>-1\land m_{2}>-1}$ is necessary and sufficient for p to be a proper probability density function.

The Pearson type II distribution

The Pearson type II distribution is a special case of the Pearson type I family restricted to symmetric distributions.

For the Pearson Type II Curve [3] ,

${\displaystyle y=y_{0}\left(1-{\frac {x^{2}}{a^{2}}}\right)^{m}}$

where

${\displaystyle x=\sum d^{2}/2-(n^{3}-n)/12}$

the ordinate, y, is the frequency of ${\displaystyle \sum d^{2}}$. The Pearson Type II Curve 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). After that, the distribution mimics a standard Student's t-distribution. For the table of values, certain values are used as the constants in the previous equation:

${\displaystyle m={\frac {5\beta _{2}-9}{2(3-\beta _{2})}}}$
${\displaystyle a^{2}={\frac {2\mu _{2}\beta _{2}}{3-\beta _{2}}}}$
${\displaystyle y_{0}={\frac {N[\Gamma (2m+2)]}{a[2^{2m+1}][\Gamma (m+1)]}}}$

The moments of x used are

${\displaystyle \mu _{2}=(n-1)[(n^{2}+n)/12]^{2}}$
${\displaystyle \beta _{2}={\frac {3(25n^{4}-13n^{3}-73n^{2}+37n+72)}{25n(n+1)^{2}(n-1)}}}$

The Pearson type III distribution

${\displaystyle \lambda =\mu 1+{\frac {b_{0}}{b_{1}}}-(m+1)b_{1}\!}$
${\displaystyle b_{0}+b_{1}(x-\lambda )\!}$ follows a :${\displaystyle gamma(m+1,b_{1}^{2})\!}$

Pearson type III distribution gamma distribution, chi-square distribution

The Pearson type V distribution

${\displaystyle \lambda =\mu _{1}-{\frac {a-C_{1}}{1-2b_{2}}}\!}$
${\displaystyle x-\lambda \!}$ follows a :${\displaystyle inversegamma({\frac {1}{b_{2}}}-1,{\frac {a-C_{1}}{b_{2}}})\!}$

Pearson type V distribution inverse-gamma distribution

The Pearson type VI distribution

${\displaystyle \lambda =\mu _{1}+(a_{2}-a_{1}){\frac {m_{2}+1}{m_{2}+m_{1}+2}}-a_{2}\!}$
${\displaystyle {\frac {x-\lambda -a_{2}}{a_{2}-a_{1}}}\!}$ follows a :${\displaystyle betaprime(m_{2}+1,-m_{2}-m_{1}-1)\!}$

Pearson type VI distribution beta prime distribution, F-distribution

Relation to other distributions

The Pearson family subsumes the following distributions, among others:

Applications

These models are used in financial markets, given their ability to be parametrised 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. and this family of distributions may prove to be one of the more important.

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

Notes

1. Miller, Jeff (2006-07-09). "Beta distribution". Earliest Known Uses of Some of the Words of Mathematics. Unknown parameter |accessyear= ignored (|access-date= suggested) (help); Unknown parameter |accessmonthday= ignored (help); Unknown parameter |coauthors= ignored (help); External link in |work= (help)
2. Miller, Jeff (2006-12-07). "Gamma distribution". Earliest Known Uses of Some of the Words of Mathematics. Unknown parameter |accessyear= ignored (|access-date= suggested) (help); Unknown parameter |accessmonthday= ignored (help); Unknown parameter |coauthors= ignored (help); External link in |work= (help)
3. Ramsey, Philip H. (1989-09-01). "Critical Values for Spearman's Rank Order Correlation". Unknown parameter |accessyear= ignored (|access-date= suggested) (help); Unknown parameter |accessmonthday= ignored (help)