The th -statistic  is the unique symmetric unbiased estimator of the cumulant  of a given statistical distribution, i.e.,  is defined so that

(1)

where  denotes the expectation value of  (Kenney and Keeping 1951, p. 189; Rose and Smith 2002, p. 256). In addition, the variance

(2)

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)

then

			(4)
			(5)
			(6)
			(7)

(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)
			(9)
			(10)
			(11)

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)
			(13)
			(14)
			(15)

An unbiased estimator for  is given by

(16)

(Kenney and Keeping 1951, p. 189). In the special case of a normal parent population, an unbiased estimator for  is given by

(17)

(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)
			(19)
			(20)
			(21)

(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.

REFERENCES:

Carver, H. C. (Ed.). "Fundamentals of the Theory of Sampling." Ann. Math. Stat. 1, 101-121, 1930.

Church, A. E. R. "On the Means and Squared Standard-Deviations of Small Samples from Any Population." Biometrika 18, 321-394, 1926.

Fisher, R. A. "Moments and Product Moments of Sampling Distributions." Proc. London Math. Soc. 30, 199-238, 1928.

Halmos, P. R. "The Theory of Unbiased Estimation." Ann. Math. Stat. 17, 34-43, 1946.

Irwin, J. O. and Kendall, M. G. "Sampling Moments of Moments for a Finite Population." Ann. Eugenics 12, 138-142, 1944.

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.

Kenney, J. F. and Keeping, E. S. "The -Statistics." §7.9 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 99-100, 1962.

Rose, C. and Smith, M. D. "k-Statistics: Unbiased Estimators of Cumulants." §7.2C in Mathematical Statistics with Mathematica. New York: Springer-Verlag, pp. 256-259, 2002.

Stuart, A.; and Ord, J. K. Kendall's Advanced Theory of Statistics, Vol. 2A: Classical Inference & the Linear Model, 6th ed. New York: Oxford University Press, 1999.

Ziaud-Din, M. "Expression of the k-Statistics  and  in Terms of Power Sums and Sample Moments." Ann. Math. Stat. 25, 800-803, 1954.

Ziaud-Din, M. "The Expression of -Statistic  in Terms of Power Sums and Sample Moments." Ann. Math. Stat. 30, 825-828, 1959.

Ziaud-Din, M. and Ahmad, M. "On the Expression of the -Statistic  in Terms of Power Sums and Sample Moments." Bull. Internat. Stat. Inst. 38, 635-640, 1960.