| 释义 |
k-StatisticAn Unbiased Estimator of the Cumulants  of a Distribution. The expectation values of the  -statistics are therefore given by the correspondingCumulants
(Kenney and Keeping 1951, p. 189). For a sample of size,  , the first few -statistics are given by
where  is the sample Mean,  is the sample Variance, and  is the sample  th Moment about theMean (Kenney and Keeping 1951, pp. 109-110, 163-165, and 189; Kenney and Keeping 1962). These statistics are obtainedfrom inverting the relationships
The first moment (sample Mean) is
 | (14) |
 | (15) |
and the expectation value is
since there are  terms  , using
 | (18) |
 is the Moment about 0. Using the identity
 | (19) |
 about the Mean and simplifying then gives
 | (20) |
 is known as Bessel's Correction.
The third Moment is
Now use the identities | (22) |  | (23) |
where it is understood that sums over products of variables exclude equal indices. Plugging in
 | (24) |
 | (25) |
 is the Moment about 0. Plugging in the identities
and simplifying then gives
 | (28) |
The fourth Moment is
Now use the identitiesPlugging in,
The expectation value is then given by
where  are Moments about 0. Using the identities
and simplifying gives
 | (38) |
The square of the second moment is
Now use the identitiesPlugging in,
The expectation value is then given by
where are Moments about 0. Using the identities
and simplifying gives
 | (48) |
The Variance of  is given by
 | (49) |
 is given by
 | (50) |
 can be expressed in terms of Cumulants by
 | (51) |
 is
 | (52) |
Now consider a finite population. Let a sample of be taken from a population of  . Then UnbiasedEstimators  for the population Mean  , for the population Variance , for the population Skewness  , and  for the population Kurtosis  are
(Church 1926, p. 357; Carver 1930; Irwin and Kendall 1944; Kenney and Keeping 1951, p. 143), where is the sampleSkewness and is the sample Kurtosis.See also Gaussian Distribution, Kurtosis, Mean, Moment, Skewness, Variance
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.
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