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.
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(5) |
giving
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(6) |
|
(7) |
Now define the raw th moment by
|
(8) |
so combining (7) with (8) gives
|
(9) |
For ,
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(10) |
so
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(11) |
and for ,
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(12) |
so
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(13) |
Combining (11), (13), and the definitions
obtained by letting and solving simultaneously gives and . Writing
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(16) |
then allows the general recurrence to be written
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(17) |
For the special cases and , this gives
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(18) |
|
(19) |
so the skewness and kurtosis excess are
The parameters , , and can therefore be written
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(22) | |||
|
(23) | |||
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(24) |
where
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(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.
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Cite this as:
Weisstein, Eric W. "Pearson System." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PearsonSystem.html