Pearson System

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

(1)

which has solution

(2)

to

(3)

which has solution

(4)

Let ,  be the roots of . Then the possible types of curves are

0. , . E.g., normal distribution.

I. , . E.g., beta distribution.

II. , ,  where .

III. , ,  where . E.g., gamma distribution. This case is intermediate to cases I and VI.

IV. , .

V. ,  where . Intermediate to cases IV and VI.

VI. ,  where  is the larger root. E.g., beta prime distribution.

VII. , , . 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 . Let  at  and , where these may be  or . If  also vanishes at , , then the th moment and th moments exist.

(5)

giving

(6)

(7)

Now define the raw th moment by

(8)

so combining (7) with (8) gives

(9)

For ,

(10)

so

(11)

and for ,

(12)

so

(13)

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

			(14)
			(15)

obtained by letting  and solving simultaneously gives  and . Writing

(16)

then allows the general recurrence to be written

(17)

For the special cases  and , this gives

(18)

(19)

so the skewness and kurtosis excess are

			(20)
			(21)

The parameters , , and  can therefore be written

			(22)
			(23)
			(24)

where

(25)

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.