info prev up next book cdrom email home

Pearson System

Generalizes the differential equation for the Gaussian Distribution

\begin{displaymath}
{dy\over dx} = {y(m-x)\over a}
\end{displaymath} (1)

to
\begin{displaymath}
{dy\over dx} = {y(m-x)\over a+bx+cx^2}.
\end{displaymath} (2)

Let $c_1$, $c_2$ be the roots of $a+bx+cx^2$. Then the possible types of curves are
0. $b=c=0$, $a>0$. E.g., Normal Distribution.

I. $b^2/4ac<0$, $c_1\leq x\leq c_2$. E.g., Beta Distribution.

II. $b^2/4ac=0$, $c<0$, $-c_1\leq x\leq c_1$ where $c_1\equiv \sqrt{-c/a}$.

III. $b^2/4ac=\infty$, $c=0$, $c_1\leq x<\infty$ where $c_1\equiv -a/b$. E.g., Gamma Distribution. This case is intermediate to cases I and VI.

IV. $0<b^2/4ac<1$, $-\infty<x<\infty$.

V. $b^2/4ac=1$, $c_1\leq x<\infty$ where $c_1\equiv -b/2a$. Intermediate to cases IV and VI.

VI. $b^2/4ac>1$, $c_1\leq x<\infty$ where $c_1$ is the larger root. E.g., Beta Prime Distribution.

VII. $b^2/4ac=0$, $c>0$, $-\infty<x<\infty$. E.g., Student's t-Distribution.

Classes IX-XII are discussed in Pearson (1916). See also Craig (in Kenney and Keeping 1951). If a Pearson curve possesses a Mode, it will be at $x=m$. Let $y(x)=0$ at $c_1$ and $c_2$, where these may be $-\infty$ or $\infty$. If $yx^{r+2}$ also vanishes at $c_1$, $c_2$, then the $r$th Moment and $(r+1)$th Moments exist.
$ \int_{c_1}^{c_2} {dy\over dx} (ax^r+bx^{r+1}+cx^{r+2})\,dx =\int_{c_1}^{c_2} y(mx^r-x^{r+1})\,dx$

(3)
giving

$[y(ax^r+bx^{r+1}+cx^{r+2})]_{c_1}^{c_2}-\int_{c_1}^{c_2} y[arx^{r-1}+b(r+1)x^r+c(r+2)x^{r+1}]\,dx$
$ = \int_{c_1}^{c_2} y(mx^r-x^{r+1})\,dx\quad$ (4)

\begin{displaymath}
0-\int_{c_1}^{c_2} y[arx^{r-1}+b(r+1)x^r+c(r+2)x^{r+1}]\,dx = \int_{c_1}^{c_2} y(mx^r-x^{r+1})\,dx
\end{displaymath} (5)

also,
\begin{displaymath}
\nu_r=\int_{c_1}^{c_2} yx^r\,dx,
\end{displaymath} (6)

so
\begin{displaymath}
ar\nu_{r-1}+b(r+1)\nu_r+c(r+2)\nu_{r+1}=-m\nu_r+\nu_{r+1}.
\end{displaymath} (7)

For $r=0$,
\begin{displaymath}
b+2c\nu_1=-m+\nu_1,
\end{displaymath} (8)

so
\begin{displaymath}
\nu_1={m+b\over 1-2c}.
\end{displaymath} (9)

For $r=1$,
\begin{displaymath}
a+2b\nu_1+3c\nu_2=-m\nu_1+\nu_2,
\end{displaymath} (10)

so
\begin{displaymath}
\nu_2={a+(m+2b)\nu_1\over 1-3c}.
\end{displaymath} (11)

Now let $t\equiv (x-\nu_1)/\sigma$. Then
$\displaystyle \nu_1$ $\textstyle =$ $\displaystyle 0$ (12)
$\displaystyle \nu_2$ $\textstyle =$ $\displaystyle \mu_2=1$ (13)
$\displaystyle \alpha_r$ $\textstyle =$ $\displaystyle \mu_r=\nu_r.$ (14)

Hence $b=-m$, and $a=1-c$ so
\begin{displaymath}
(1-3c)r\alpha_{r-1}-mr\alpha_r+[c(r+2)-1]\alpha_{r+1}=0.
\end{displaymath} (15)

For $r=2$,
\begin{displaymath}
2m+(1-4c)\alpha_3=0.
\end{displaymath} (16)

For $r=3$,
\begin{displaymath}
3(1-3c)-3m\alpha_3-(1-5c)\alpha_4=0.
\end{displaymath} (17)

So the Skewness and Kurtosis are
$\displaystyle \gamma_1$ $\textstyle =$ $\displaystyle \alpha_3={2m\over 4c-1}$ (18)
$\displaystyle \gamma_2$ $\textstyle =$ $\displaystyle \alpha_4-3 = {6(m^2-4c^2+c)\over (4c-1)(5c-1)}.$ (19)

So the parameters $a$, $b$, and $c$ can be written
$\displaystyle a$ $\textstyle =$ $\displaystyle 1-3c$ (20)
$\displaystyle b$ $\textstyle =$ $\displaystyle -m={\gamma_1\over 2(1+2\delta)}$ (21)
$\displaystyle c$ $\textstyle =$ $\displaystyle {\delta\over 2(1+2\delta)},$ (22)

where
\begin{displaymath}
\delta\equiv {2\gamma_2-3{\gamma_1}^2\over \gamma_2+6}.
\end{displaymath} (23)


References

Craig, C. C. ``A New Exposition and Chart for the Pearson System of Frequency Curves.'' Ann. Math. Stat. 7, 16-28, 1936.

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 107, 1951.

Pearson, K. ``Second Supplement to a Memoir on Skew Variation.'' Phil. Trans. A 216, 429-457, 1916.



info prev up next book cdrom email home

© 1996-9 Eric W. Weisstein
1999-05-26