The -statistic is the unique symmetric unbiased estimator for a central moment of a distribution
(1)
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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. 254). The first few are given in terms of power sums by
(3)
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(4)
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(5)
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(6)
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and in terms of sample central moments by
(7)
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(8)
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(9)
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(10)
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These can be given by HStatistic[r] and HStatisticToSampleCentral[r], respectively, in the Wolfram Language application package mathStatica.