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

(1) |

(2) |

(3) |

(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.

**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.

© 1996-9

1999-05-25