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


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


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 k_n for k-statistics, while Rose and Smith (2002) prefer k_n.

The k-statistics can be given in terms of the sums of the rth powers of the data points as




(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 n, the first few k-statistics are given by


where mu is the sample mean, m_2 is the sample variance, and m_i is the ith sample central moment (Kenney and Keeping 1951, pp. 109-110, 163-165, and 189; Kenney and Keeping 1962).

The variances of the first few k-statistics are given by


An unbiased estimator for var(k_2) is given by


(Kenney and Keeping 1951, p. 189). In the special case of a normal parent population, an unbiased estimator for var(k_3) is given by


(Kenney and Keeping 1951, pp. 189-190).

For a finite population, let a sample size n be taken from a population size N. Then unbiased estimators M_1 for the population mean mu, M_2 for the population variance mu_2, G_1 for the population skewness gamma_1, and G_2 for the population kurtosis excess gamma_2 are


(Church 1926, p. 357; Carver 1930; Irwin and Kendall 1944; Kenney and Keeping 1951, p. 143), where gamma_1 is the sample skewness and gamma_2 is the sample kurtosis excess.

See also

Cumulant, h-Statistic, Kurtosis, Mean, Moment, Normal Distribution, Polykay, Sample Central Moment, Skewness, Statistic, Unbiased Estimator, Variance

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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 k-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 k_9 and k_(10) in Terms of Power Sums and Sample Moments." Ann. Math. Stat. 25, 800-803, 1954.Ziaud-Din, M. "The Expression of k-Statistic k_(11) 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 k-Statistic k_(12) in Terms of Power Sums and Sample Moments." Bull. Internat. Stat. Inst. 38, 635-640, 1960.

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Cite this as:

Weisstein, Eric W. "k-Statistic." From MathWorld--A Wolfram Web Resource.

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