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 =

(3)

h_2 = [画像:(nS_2-S_1^2)/((n-1)n)]

(4)

h_3 = [画像:(2S_1^3-3nS_1S_2+n^2S_3)/((n-2)(n-1)n)]

(5)

h_4 = [画像:(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 =

(7)

h_2 = (nm_2)/(n-1)

(8)

h_3 = [画像:(n^2m_3)/((n-2)(n-1))]

(9)

h_4 = [画像:(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.

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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

h-Statistic

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