h-Statistic

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

(1)

In addition, the variance

(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

			(3)
			(4)
			(5)
			(6)

and in terms of sample central moments by

			(7)
			(8)
			(9)
			(10)

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

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.