Cumulant
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover,
الجزء والصفحة:
...
19-2-2021
1539
Cumulant
Let 
be the characteristic function, defined as the Fourier transform of the probability density function 
using Fourier transform parameters 
,
The cumulants 
are then defined by
 |
(3)
|
(Abramowitz and Stegun 1972, p. 928). Taking the Maclaurin series gives
 |
(4)
|
where 
are raw moments, so
These transformations can be given by CumulantToRaw[n] in the Mathematica application package mathStatica.
In terms of the central moments 
,
where 
is the mean and 
is the variance. These transformations can be given by CumulantToCentral[n].
Multivariate cumulants can be expressed in terms of raw moments, e.g.,
and central moments, e.g.,
using CumulantToRaw[![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/Cumulant/Inline66.gif" style="height:15px; width:5px" />m, n, ...![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/Cumulant/Inline67.gif" style="height:15px; width:5px" />] and CumulantToCentral[![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/Cumulant/Inline68.gif" style="height:15px; width:5px" />m, n, ...![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/Cumulant/Inline69.gif" style="height:15px; width:5px" />], respectively.
The k-statistics are unbiased estimators of the cumulants.
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 928, 1972.
Kenney, J. F. and Keeping, E. S. "Cumulants and the Cumulant-Generating Function," "Additive Property of Cumulants," and "Sheppard's Correction." §4.10-4.12 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 77-82, 1951.
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