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

(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.

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A. E. R. "On the Means and Squared Standard-Deviations of Small Samples
from Any Population." Biometrika18, 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. Eugenics12, 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.