Let be the characteristic function, defined as the Fourier transform of the probability density function using Fourier transform parameters ,
(1)
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(2)
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The cumulants are then defined by
(3)
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(Abramowitz and Stegun 1972, p. 928). Taking the Maclaurin series gives
(4)
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where are raw moments, so
(5)
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(6)
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(7)
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(8)
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(9)
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These transformations can be given by CumulantToRaw[n] in the Mathematica application package mathStatica.
In terms of the central moments ,
(10)
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(11)
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(12)
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(13)
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(14)
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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.,
(15)
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(16)
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and central moments, e.g.,
(17)
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(18)
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(19)
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(20)
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(21)
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using CumulantToRaw[m, n, ...] and CumulantToCentral[m, n, ...], respectively.
The k-statistics are unbiased estimators of the cumulants.