Pearson System

Generalizes the differential equation for the Gaussian Distribution

$${dy\over dx} = {y(m-x)\over a}$$ (1)

to

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

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

also,

$$\nu_r=\int_{c_1}^{c_2} yx^r\,dx,$$ (6)

so

$$ar\nu_{r-1}+b(r+1)\nu_r+c(r+2)\nu_{r+1}=-m\nu_r+\nu_{r+1}.$$ (7)

For $r=0$,

$$b+2c\nu_1=-m+\nu_1,$$ (8)

so

$$\nu_1={m+b\over 1-2c}.$$ (9)

For $r=1$,

$$a+2b\nu_1+3c\nu_2=-m\nu_1+\nu_2,$$ (10)

so

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

Now let $t\equiv (x-\nu_1)/\sigma$. Then

$$\nu_1 = 0$$ (12)

$$\nu_2 = \mu_2=1$$ (13)

$$\alpha_r = \mu_r=\nu_r.$$ (14)

Hence $b=-m$, and $a=1-c$ so

$$(1-3c)r\alpha_{r-1}-mr\alpha_r+[c(r+2)-1]\alpha_{r+1}=0.$$ (15)

For $r=2$,

$$2m+(1-4c)\alpha_3=0.$$ (16)

For $r=3$,

$$3(1-3c)-3m\alpha_3-(1-5c)\alpha_4=0.$$ (17)

So the Skewness and Kurtosis are

$$\gamma_1 = \alpha_3={2m\over 4c-1}$$ (18)

$$\gamma_2 = \alpha_4-3 = {6(m^2-4c^2+c)\over (4c-1)(5c-1)}.$$ (19)

So the parameters $a$, $b$, and $c$ can be written

$$a = 1-3c$$ (20)

$$b = -m={\gamma_1\over 2(1+2\delta)}$$ (21)

$$c = {\delta\over 2(1+2\delta)},$$ (22)

where

$$\delta\equiv {2\gamma_2-3{\gamma_1}^2\over \gamma_2+6}.$$ (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.