The th -statistic is the unique symmetric unbiased estimator of the cumulant of a given statistical distribution, i.e., is defined so that
(1)
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where denotes the expectation value of (Kenney and Keeping 1951, p. 189; Rose and Smith 2002, p. 256). In addition, the variance
(2)
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is a minimum compared to all other unbiased estimators (Halmos 1946; Rose and Smith 2002, p. 256). Most authors (e.g., Kenney and Keeping 1951, 1962) use the notation for -statistics, while Rose and Smith (2002) prefer .
The -statistics can be given in terms of the sums of the th powers of the data points as
(3)
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then
(4)
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(5)
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(6)
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(7)
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(Fisher 1928; Rose and Smith 2002, p. 256). These can be given by KStatistic[r] in the Mathematica application package mathStatica.
For a sample size , the first few -statistics are given by
(8)
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(9)
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(10)
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(11)
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where is the sample mean, is the sample variance, and is the th sample central moment (Kenney and Keeping 1951, pp. 109-110, 163-165, and 189; Kenney and Keeping 1962).
The variances of the first few -statistics are given by
(12)
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(13)
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(14)
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(15)
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An unbiased estimator for is given by
(16)
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(Kenney and Keeping 1951, p. 189). In the special case of a normal parent population, an unbiased estimator for is given by
(17)
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(Kenney and Keeping 1951, pp. 189-190).
For a finite population, let a sample size be taken from a population size . Then unbiased estimators for the population mean , for the population variance , for the population skewness , and for the population kurtosis excess are
(18)
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(19)
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(20)
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(21)
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(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 excess.