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h-Statistic

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The h-statistic h_r is the unique symmetric unbiased estimator for a central moment of a distribution

<h_r>=mu_r.
(1)

In addition, the variance

var(h_r)=<(h_r-mu_r)^2>
(2)

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

h_1=0
(3)
h_2=(nS_2-S_1^2)/((n-1)n)
(4)
h_3=(2S_1^3-3nS_1S_2+n^2S_3)/((n-2)(n-1)n)
(5)
h_4=(6nS_1^2S_2+3(3-2n)S_2^2-4(n^2-2n+3)S_1S_3+(n^3-2n^2+3n)S_4-3S_1^4)/((n-3)(n-2)(n-1)n),
(6)

and in terms of sample central moments by

h_1=0
(7)
h_2=(nm_2)/(n-1)
(8)
h_3=(n^2m_3)/((n-2)(n-1))
(9)
h_4=(3(3-2n)n^2m_2^2+n^2(n^2-2n+3)m_4)/((n-3)(n-2)(n-1)n).
(10)

These can be given by HStatistic[r] and HStatisticToSampleCentral[r], respectively, in the Wolfram Language application package mathStatica.

See also

Central Moment, k-Statistic, Polyache

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References

Dwyer, P. S. "Moments of Any Rational Integral Isobaric Sample Moment Function." Ann. Math. Stat. 8, 21-65, 1937.Halmos, P. R. "The Theory of Unbiased Estimation." Ann. Math. Stat. 17, 34-43, 1946.Rose, C. and Smith, M. D. "h-Statistics: Unbiased Estimators of Central Moments." §7.2B in Mathematical Statistics with Mathematica. New York: Springer-Verlag, pp. 253-256, 2002.

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h-Statistic

Cite this as:

Weisstein, Eric W. "h-Statistic." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/h-Statistic.html

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