Cumulant
Let 
be the Characteristic Function, defined as the Fourier Transform of the Probability Density Function,
 | (1) |
are defined by | (2) |
 | |
 | |
 | |
 | |
 | (3) |
are Moments about 0, so |  |  | (4) |
 | |  | (5) |
 | |  | (6) |
 | |  | (7) |
 | |  | (8) |
In terms of the Moments
about the Mean, | |  | (9) |
| |  | (10) |
| |  | (11) |
| |  | (12) |
| |  | (13) |
where
is the Mean and 
is the Variance.The k-Statistic are Unbiased Estimators of the cumulants.
See also Characteristic Function, Cumulant-Generating Function, k-Statistic, Kurtosis,Mean, Moment, Sheppard's Correction, Skewness, Variance
References
Abramowitz, M. and Stegun, C. 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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