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Pearson System

A system of equation types obtained by generalizing the differential equation for the normal distribution

(dy)/(dx)=(y(m-x))/a,
(1)

which has solution

y=Ce^((2m-x)x/(2a)),
(2)

to

(dy)/(dx)=(y(m-x))/(a+bx+cx^2),
(3)

which has solution

y=C(a+bx+cx^2)^(-1/(2c))exp[((b+2cm)tan^(-1)((b+2cx)/(sqrt(4ac-b^2))))/(csqrt(4ac-b^2))].
(4)

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<=x<=c_2. E.g., beta distribution.

II. b^2/4ac=0, c<0, -c_1<=x<=c_1 where c_1=sqrt(-c/a).

III. b^2/4ac=infty, c=0, c_1<=x<infty where c_1=-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<=x<infty where c_1=-b/2a. Intermediate to cases IV and VI.

VI. b^2/4ac>1, c_1<=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 rth moment and (r+1)th moments exist.

int_(c_1)^(c_2)(dy)/(dx)(ax^r+bx^(r+1)+cx^(r+2))dx=int_(c_1)^(c_2)y(mx^r-x^(r+1))dx,
(5)

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
(6)
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.
(7)

Now define the raw rth moment by

nu_r=int_(c_1)^(c_2)yx^rdx,
(8)

so combining (7) with (8) gives

arnu_(r-1)+b(r+1)nu_r+c(r+2)nu_(r+1)=-mnu_r+nu_(r+1).
(9)

For r=0,

b+2cnu_1=-m+nu_1,
(10)

so

nu_1=(m+b)/(1-2c),
(11)

and for r=1,

a+2bnu_1+3cnu_2=-mnu_1+nu_2,
(12)

so

nu_2=(a+(m+2b)nu_1)/(1-3c).
(13)

Combining (11), (13), and the definitions

nu_1=0
(14)
nu_2=mu_2=1
(15)

obtained by letting t=(x-nu_1)/sigma and solving simultaneously gives b=-m and a=1-3c. Writing

alpha_r=mu_r=nu_r
(16)

then allows the general recurrence to be written

(1-3c)ralpha_(r-1)-mralpha_r+[c(r+2)-1]alpha_(r+1)=0.
(17)

For the special cases r=2 and r=3, this gives

2m+(1-4c)alpha_3=0
(18)
3(1-3c)-3malpha_3-(1-5c)alpha_4=0,
(19)

so the skewness and kurtosis excess are

gamma_1=alpha_3=(2m)/(4c-1)
(20)
gamma_2=alpha_4-3=(6(m^2-4c^2+c))/((4c-1)(5c-1)).
(21)

The parameters a, b, and c can therefore be written

a=1-3c
(22)
b=-m=(gamma_1)/(2(1+2delta))
(23)
c=delta/(2(1+2delta)),
(24)

where

delta=(2gamma_2-3gamma_1^2)/(gamma_2+6).
(25)

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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.

Referenced on Wolfram|Alpha

Pearson System

Cite this as:

Weisstein, Eric W. "Pearson System." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PearsonSystem.html

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