Pearson distribution

Diagram of the Pearson system, showing distributions of types I, III, VI, V, and IV in terms of β₁(squared skewness) and β₂(traditional kurtosis)

ThePearson distributionis a family ofcontinuousprobability distributions.It was first published byKarl Pearsonin 1895 and subsequently extended by him in 1901 and 1916 in a series of articles onbiostatistics.

History

The Pearson system was originally devised in an effort to model visiblyskewedobservations. It was well known at the time how to adjust a theoretical model to fit the first twocumulantsormomentsof observed data: Anyprobability distributioncan be extended straightforwardly to form alocation-scale family.Except inpathologicalcases, a location-scale family can be made to fit the observedmean(first cumulant) andvariance(second cumulant) arbitrarily well. However, it was not known how to construct probability distributions in which theskewness(standardized third cumulant) andkurtosis(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 thenormal distribution(which was originally known as type V). The classification depended on whether the distributions weresupportedon a bounded interval, on a half-line, or on the wholereal 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 thenormal distribution,but now theinverse-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 β₁and β₂.The first is the square of theskewness:β₁= γ₁where γ₁is the skewness, or thirdstandardized moment.The second is the traditionalkurtosis,or fourth standardized moment: β₂= γ₂+ 3. (Modern treatments define kurtosis γ₂in 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 on the right shows which Pearson type a given concrete distribution (identified by a point (β₁,β₂)) belongs to.

Many of the skewed and/or non-mesokurticdistributions familiar to us today were still unknown in the early 1890s. What is now known as thebeta distributionhad been used byThomas Bayesas aposterior distributionof the parameter of aBernoulli distributionin his 1763 work oninverse 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.) Thegamma distributionoriginated 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 containsStudent'st-distributionas a special case, predatingWilliam Sealy Gosset's subsequent use by several years. His 1901 paper introduced theinverse-gamma distribution(type V) and thebeta prime distribution(type VI).

Definition

A Pearsondensitypis defined to be any valid solution to thedifferential equation(cf. Pearson 1895, p. 381)

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

with:

${\begin{aligned}b_{0}&={\frac {4\beta _{2}-3\beta _{1}}{10\beta _{2}-12\beta _{1}-18}}\mu _{2},\\[5pt]a=b_{1}&={\sqrt {\mu _{2}}}{\sqrt {\beta _{1}}}{\frac {\beta _{2}+3}{10\beta _{2}-12\beta _{1}-18}},\\[5pt]b_{2}&={\frac {2\beta _{2}-3\beta _{1}-6}{10\beta _{2}-12\beta _{1}-18}}.\end{aligned}}$

According to Ord,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 thenormal distribution(which gives a linear function) and, secondly, from a recurrence relation for values in theprobability mass functionof thehypergeometric distribution(which yields the linear-divided-by-quadratic structure).

In Equation (1), the parameteradetermines astationary point,and hence under some conditions amodeof the distribution, since

$p'(\mu -a)=0$

follows directly from the differential equation.

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

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

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 thediscriminant(and hence the number of realroots) of thequadratic function

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

Particular types of distribution

Case 1, negative discriminant

The Pearson type IV distribution

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

${\begin{aligned}y&=x+{\frac {b_{1}}{2b_{2}}},\\[5pt]\alpha &={\frac {\sqrt {4b_{2}b_{0}-b_{1}^{2}}}{2b_{2}}}.\end{aligned}}$

Observe thatαis a well-defined real number andα≠ 0,because by assumption$4b_{2}b_{0}-b_{1}^{2}>0$and thereforeb₂≠ 0.Applying these substitutions, the quadratic function (2) is transformed into

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

The absence of real roots is obvious from this formulation, because α²is necessarily positive.

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

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

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

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

involves theinversetrigonometricarctan function. Then

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

Finally, let

${\begin{aligned}m&={\frac {1}{2b_{2}}},\\[5pt]\nu &=-{\frac {2b_{2}a-b_{1}}{2b_{2}^{2}\alpha }}.\end{aligned}}$

Applying these substitutions, we obtain the parametric function:

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

This unnormalized density hassupporton the entirereal line.It depends on ascale parameterα > 0 andshape parametersm> 1/2 andν.One parameter was lost when we chose to find the solution to the differential equation (1) as a function ofyrather thanx.We therefore reintroduce a fourth parameter, namely thelocation parameterλ.We have thus derived the density of thePearson type IV distribution:

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

Thenormalizing constantinvolves thecomplexGamma function(Γ) and theBeta function(B). Notice that thelocation parameterλhere is not the same as the original location parameter introduced in the general formulation, but is related via

$\lambda =\lambda _{original}+{\frac {\alpha \nu }{2(m-1)}}.$

The Pearson type VII distribution

Plot of Pearson type VII densities withλ= 0,σ= 1, and:γ₂= ∞ (red);γ₂= 4 (blue); andγ₂= 0 (black)

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

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

where B is theBeta function.

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

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

which requiresm> 3/2. This entails a minor loss of generality but ensures that thevarianceof the distribution exists and is equal to σ².Now the parametermonly controls thekurtosisof the distribution. Ifmapproaches infinity asλandσare held constant, thenormal distributionarises as a special case:

${\begin{aligned}&\lim _{m\to \infty }{\frac {1}{\sigma {\sqrt {2m-3}}\,\operatorname {\mathrm {B} } \left(m-{\frac {1}{2}},{\frac {1}{2}}\right)}}\left[1+\left({\frac {x-\lambda }{\sigma {\sqrt {2m-3}}}}\right)^{2}\right]^{-m}\\[5pt]={}&{\frac {1}{\sigma {\sqrt {2}}\,\operatorname {\Gamma } \left({\frac {1}{2}}\right)}}\cdot \lim _{m\to \infty }{\frac {\Gamma (m)}{\operatorname {\Gamma } \left(m-{\frac {1}{2}}\right){\sqrt {m-{\frac {3}{2}}}}}}\cdot \lim _{m\to \infty }\left[1+{\frac {\left({\frac {x-\lambda }{\sigma }}\right)^{2}}{2m-3}}\right]^{-m}\\[5pt]={}&{\frac {1}{\sigma {\sqrt {2\pi }}}}\cdot 1\cdot \exp \left[-{\frac {1}{2}}\left({\frac {x-\lambda }{\sigma }}\right)^{2}\right].\end{aligned}}$

This is the density of a normal distribution with meanλand standard deviationσ.

It is convenient to require thatm> 5/2 and to let

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

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 (λ, σ, γ₂) has a mean ofλ,standard deviationofσ,skewnessof zero, and positiveexcess kurtosisof γ₂.

Student'st-distribution

The Pearson type VII distribution is equivalent to the non-standardizedStudent'st-distributionwith parameters ν > 0, μ, σ²by applying the following substitutions to its original parameterization:

${\begin{aligned}\lambda &=\mu,\\[5pt]\alpha &={\sqrt {\nu \sigma ^{2}}},\\[5pt]m&={\frac {\nu +1}{2}},\end{aligned}}$

Observe that the constraintm> 1/2is satisfied.

The resulting density is

$p(x\mid \mu,\sigma ^{2},\nu )={\frac {1}{{\sqrt {\nu \sigma ^{2}}}\,\operatorname {\mathrm {B} } \left({\frac {\nu }{2}},{\frac {1}{2}}\right)}}\left(1+{\frac {1}{\nu }}{\frac {(x-\mu )^{2}}{\sigma ^{2}}}\right)^{-{\frac {\nu +1}{2}}},$

which is easily recognized as the density of a Student'st-distribution.

This implies that the Pearson type VII distribution subsumes the standardStudent'st-distributionand also the standardCauchy distribution.In particular, the standard Student'st-distribution arises as a subcase, whenμ= 0 andσ²= 1, equivalent to the following substitutions:

${\begin{aligned}\lambda &=0,\\[5pt]\alpha &={\sqrt {\nu }},\\[5pt]m&={\frac {\nu +1}{2}},\end{aligned}}$

The density of this restricted one-parameter family is a standard Student'st:

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

Case 2, non-negative discriminant

If the quadratic function (2) has a non-negative discriminant ($b_{1}^{2}-4b_{2}b_{0}\geq 0$), it has real rootsa₁anda₂(not necessarily distinct):

${\begin{aligned}a_{1}&={\frac {-b_{1}-{\sqrt {b_{1}^{2}-4b_{2}b_{0}}}}{2b_{2}}},\\[5pt]a_{2}&={\frac {-b_{1}+{\sqrt {b_{1}^{2}-4b_{2}b_{0}}}}{2b_{2}}}.\end{aligned}}$

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

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

and the solution to the differential equation is therefore

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

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

$\int {\frac {x-a}{(x-a_{1})(x-a_{2})}}\,dx={\frac {(a_{1}-a)\ln(x-a_{1})-(a_{2}-a)\ln(x-a_{2})}{a_{1}-a_{2}}}+C$

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

Using the substitution

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

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

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

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

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

The Pearson type I distribution

ThePearson type I distribution(a generalization of thebeta distribution) arises when the roots of the quadratic equation (2) are of opposite sign, that is,$a_{1}<0<a_{2}$.Then the solutionpis supported on the interval$(a_{1},a_{2})$.Apply the substitution

$x=a_{1}+y(a_{2}-a_{1}),$

where$0<y<1$,which yields a solution in terms ofythat is supported on the interval (0, 1):

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

One may define:

${\begin{aligned}m_{1}&={\frac {a-a_{1}}{b_{2}(a_{1}-a_{2})}},\\[5pt]m_{2}&={\frac {a-a_{2}}{b_{2}(a_{2}-a_{1})}}.\end{aligned}}$

Regrouping constants and parameters, this simplifies to:

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

Thus${\frac {x-\lambda -a_{1}}{a_{2}-a_{1}}}$follows a$\mathrm {B} (m_{1}+1,m_{2}+1)$with$\lambda =\mu _{1}-(a_{2}-a_{1}){\frac {m_{1}+1}{m_{1}+m_{2}+2}}-a_{1}$.It turns out thatm₁,m₂> −1 is necessary and sufficient forpto be a proper probability density function.

The Pearson type II distribution

ThePearson type II distributionis a special case of the Pearson type I family restricted to symmetric distributions.

For the Pearson type II curve,

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

where

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

The ordinate,y,is the frequency of$\sum d^{2}$.The Pearson type II distribution is used in computing the table of significant correlation coefficients forSpearman's rank correlation coefficientwhen the number of items in a series is less than 100 (or 30, depending on some sources). After that, the distribution mimics a standardStudent's t-distribution.For the table of values, certain values are used as the constants in the previous equation:

${\begin{aligned}m&={\frac {5\beta _{2}-9}{2(3-\beta _{2})}},\\[5pt]a^{2}&={\frac {2\mu _{2}\beta _{2}}{3-\beta _{2}}},\\[5pt]y_{0}&={\frac {N[\Gamma (2m+2)]}{a[2^{2m+1}][\Gamma (m+1)]}}.\end{aligned}}$

The moments ofxused are

${\begin{aligned}\mu _{2}&=(n-1)[(n^{2}+n)/12]^{2},\\[5pt]\beta _{2}&={\frac {3(25n^{4}-13n^{3}-73n^{2}+37n+72)}{25n(n+1)^{2}(n-1)}}.\end{aligned}}$

The Pearson type III distribution

Defining

$\lambda =\mu _{1}+{\frac {b_{0}}{b_{1}}}-(m+1)b_{1},$

$b_{0}+b_{1}(x-\lambda )$is$\operatorname {Gamma} (m+1,b_{1}^{2})$.The Pearson type III distribution is agamma distributionorchi-squared distribution.

The Pearson type V distribution

Defining new parameters:

${\begin{aligned}C_{1}&={\frac {b_{1}}{2b_{2}}},\\\lambda &=\mu _{1}-{\frac {a-C_{1}}{1-2b_{2}}},\end{aligned}}$

$x-\lambda$follows an$\operatorname {InverseGamma} ({\frac {1}{b_{2}}}-1,{\frac {a-C_{1}}{b_{2}}})$.The Pearson type V distribution is aninverse-gamma distribution.

The Pearson type VI distribution

Defining

$\lambda =\mu _{1}+(a_{2}-a_{1}){\frac {m_{2}+1}{m_{2}+m_{1}+2}}-a_{2},$

${\frac {x-\lambda -a_{2}}{a_{2}-a_{1}}}$follows a$\beta ^{\prime }(m_{2}+1,-m_{2}-m_{1}-1)$.The Pearson type VI distribution is abeta prime distributionorF-distribution.

Relation to other distributions

The Pearson family subsumes the following distributions, among others:

• Beta distribution(type I)
• Beta prime distribution(type VI)
• Cauchy distribution(type IV)
• Chi-squared distribution(type III)
• Continuous uniform distribution(limit of type I)
• Exponential distribution(type III)
• Gamma distribution(type III)
• F-distribution(type VI)
• Inverse-chi-squared distribution(type V)
• Inverse-gamma distribution(type V)
• Normal distribution(limit of type I, III, IV, V, or VI)
• Student'st-distribution(type VII, which is the non-skewed subtype of type IV)

Alternatives to the Pearson system of distributions for the purpose of fitting distributions to data are thequantile-parameterized distributions(QPDs) and themetalog distributions.QPDs and metalogs can provide greater shape and bounds flexibility than the Pearson system. Instead of fitting moments, QPDs are typically fit toempirical CDFor other data withlinear least squares.

Examples of modern alternatives to the Pearson skewness-vs-kurtosis diagram are: (i)https://github.com/SchildCode/PearsonPlotand (ii) the "Cullen and Frey graph" in the statistical application R.

Applications

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.,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.

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

Sources

Primary sources

• Pearson, Karl(1893)."Contributions to the mathematical theory of evolution [abstract]".Proceedings of the Royal Society.54(326–330): 329–333.doi:10.1098/rspl.1893.0079.JSTOR115538.

• Pearson, Karl(1895)."Contributions to the mathematical theory of evolution, II: Skew variation in homogeneous material"(PDF).Philosophical Transactions of the Royal Society.186:343–414.Bibcode:1895RSPTA.186..343P.doi:10.1098/rsta.1895.0010.JSTOR90649.

• Pearson, Karl(1901)."Mathematical contributions to the theory of evolution, X: Supplement to a memoir on skew variation".Philosophical Transactions of the Royal Society A.197(287–299): 443–459.Bibcode:1901RSPTA.197..443P.doi:10.1098/rsta.1901.0023.JSTOR90841.

• Pearson, Karl(1916)."Mathematical contributions to the theory of evolution, XIX: Second supplement to a memoir on skew variation".Philosophical Transactions of the Royal Society A.216(538–548): 429–457.Bibcode:1916RSPTA.216..429P.doi:10.1098/rsta.1916.0009.JSTOR91092.

• Rhind, A. (July–October 1909)."Tables to facilitate the computation of the probable errors of the chief constants of skew frequency distributions".Biometrika.7(1/2): 127–147.doi:10.1093/biomet/7.1-2.127.JSTOR2345367.

Secondary sources

• Milton Abramowitz and Irene A. Stegun (1964).Handbook of Mathematical Functionswith Formulas, Graphs, and Mathematical Tables.National Bureau of Standards.
• Eric W. Weissteinet al.Pearson Type III Distribution.FromMathWorld.

References

• Elderton, Sir W.P, Johnson, N.L. (1969)Systems of Frequency Curves.Cambridge University Press.
• Ord J.K. (1972)Families of Frequency Distributions.Griffin, London.