An Unbiased Estimator of the Cumulants of a Distribution. The expectation values of the -statistics are therefore given by the corresponding
Cumulants

(1) | |||

(2) | |||

(3) | |||

(4) |

(Kenney and Keeping 1951, p. 189). For a sample of size, , the first few -statistics are given by

(5) | |||

(6) | |||

(7) | |||

(8) |

where is the sample Mean, is the sample Variance, and is the sample th Moment about the Mean (Kenney and Keeping 1951, pp. 109-110, 163-165, and 189; Kenney and Keeping 1962). These statistics are obtained from inverting the relationships

(9) | |||

(10) | |||

(11) | |||

(12) | |||

(13) |

The first moment (sample Mean) is

(14) |

(15) |

(16) |

and the expectation value is

(17) |

since there are terms , using

(18) |

(19) |

(20) |

The third Moment is

(21) |

Now use the identities

(22) | |

(23) |

(24) |

(25) |

(26) | |||

(27) |

and simplifying then gives

(28) |

The fourth Moment is

(29) |

Now use the identities

(30) | |

(31) | |

(32) |

(33) |

The expectation value is then given by

(34) |

where are Moments about 0. Using the identities

(35) | |||

(36) | |||

(37) |

and simplifying gives

(38) |

The square of the second moment is

(39) |

Now use the identities

(40) | |

(41) | |

(42) |

(43) |

The expectation value is then given by

(44) |

where are Moments about 0. Using the identities

(45) | |||

(46) | |||

(47) |

and simplifying gives

(48) |

The Variance of is given by

(49) |

(50) |

(51) |

(52) |

Now consider a finite population. Let a sample of be taken from a population of . Then Unbiased Estimators for the population Mean , for the population Variance , for the population Skewness , and for the population Kurtosis are

(53) | |||

(54) | |||

(55) | |||

(56) |

(Church 1926, p. 357; Carver 1930; Irwin and Kendall 1944; Kenney and Keeping 1951, p. 143), where is the sample Skewness and is the sample Kurtosis.

**References**

Carver, H. C. (Ed.). ``Fundamentals of the Theory of Sampling.'' *Ann. Math. Stat.* **1**, 101-121, 1930.

Church, A. E. R. ``On the Means and Squared Standard-Deviations of Small Samples from Any Population.'' *Biometrika* **18**,
321-394, 1926.

Irwin, J. O. and Kendall, M. G. ``Sampling Moments of Moments for a Finite Population.'' *Ann. Eugenics* **12**, 138-142, 1944.

Kenney, J. F. and Keeping, E. S. *Mathematics of Statistics, Pt. 2, 2nd ed.* Princeton, NJ: Van Nostrand, 1951.

Kenney, J. F. and Keeping, E. S. ``The -Statistics.'' §7.9 in *Mathematics of Statistics, Pt. 1, 3rd ed.*
Princeton, NJ: Van Nostrand, pp. 99-100, 1962.

© 1996-9

1999-05-26