Pearson 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

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

Observe that α is a well-defined real number and α ≠ 0, because by assumption ${\displaystyle 4b_{2}b_{0}-b_{1}^{2}>0}$ and therefore b₂ ≠ 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 α² is necessarily positive.

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

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

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

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

involves the inverse trigonometric arctan function. Then

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

Finally, let

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

Applying these substitutions, we obtain the parametric function:

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

This unnormalized density has support on the entire real line. It depends on a scale parameter α > 0 and shape parameters m > 1/2 and ν. 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:

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

The normalizing constant involves the complex Gamma function (Γ) and the Beta function (B). Notice that the location parameter λ here is not the same as the original location parameter introduced in the general formulation, but is related via

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

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

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

where B is the Beta function.

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

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

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. 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:

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

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

It is convenient to require that 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 (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:

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

Observe that the constraint m > 1/2 is satisfied.

The resulting density is

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

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

This implies that the Pearson type VII distribution subsumes the standard Student's t-distribution and also the standard Cauchy distribution. In particular, the standard Student's t-distribution arises as a subcase, when μ = 0 and σ² = 1, equivalent to the following substitutions:

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

The density of this restricted one-parameter family is a standard Student's t:

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

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):

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

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

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

and the solution to the differential equation is therefore

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

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

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

involves only 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):

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

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

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

The Pearson type I distribution (a generalization of the beta distribution) arises when the roots of the quadratic equation (2) are of opposite sign, that is, ${\displaystyle a_{1}<0<a_{2}}$. Then the solution p is supported on the interval ${\displaystyle (a_{1},a_{2})}$. Apply the substitution

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

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

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

One may define:

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

Regrouping constants and parameters, this simplifies to:

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

Thus ${\displaystyle {\frac {x-\lambda -a_{1}}{a_{2}-a_{1}}}}$ follows a ${\displaystyle \mathrm {B} (m_{1}+1,m_{2}+1)}$ with ${\displaystyle \lambda =\mu _{1}-(a_{2}-a_{1}){\frac {m_{1}+1}{m_{1}+m_{2}+2}}-a_{1}}$. It turns out that m₁, m₂ > −1 is necessary and sufficient for p to be a proper probability density function.

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,[4]

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

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:

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

The moments of x used are

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

Defining

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

${\displaystyle b_{0}+b_{1}(x-\lambda )}$ is ${\displaystyle \operatorname {Gamma} (m+1,b_{1}^{2})}$. The Pearson type III distribution is a gamma distribution or chi-squared distribution.

Defining new parameters:

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

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

Defining

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

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

• 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. JSTOR 115538.

• 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. JSTOR 90649.

• 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. JSTOR 90841.

• 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. JSTOR 91092.

• 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. JSTOR 2345367.

• Milton Abramowitz and Irene A. Stegun (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards.
• Eric W. Weisstein et al. Pearson Type III Distribution. From MathWorld.

• 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.