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Date: 2-3-2021
1232
Date: 14-2-2016
2418
Date: 26-4-2021
1328
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Let be the characteristic function, defined as the Fourier transform of the probability density function using Fourier transform parameters ,
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(2) |
The cumulants are then defined by
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(Abramowitz and Stegun 1972, p. 928). Taking the Maclaurin series gives
(4) |
where are raw moments, so
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These transformations can be given by CumulantToRaw[n] in the Mathematica application package mathStatica.
In terms of the central moments ,
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(12) |
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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.,
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and central moments, e.g.,
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using CumulantToRaw[m, n, ...] and CumulantToCentral[m, n, ...], 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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