## Cumulant

Let be the Characteristic Function, defined as the Fourier Transform of the Probability Density Function,

 (1)

Then the cumulants are defined by
 (2)

Taking the Maclaurin Series gives
 (3)
where 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.