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k-Statistic


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

 <k_n>=kappa_n,
(1)

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

 var(k_r)=<(k_r-kappa_r)^2>
(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 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

 S_r=sum_(i=1)^nX_i^r,
(3)

then

k_1=(S_1)/n
(4)
k_2=(nS_2-S_1^2)/(n(n-1))
(5)
k_3=(2S_1^3-3nS_1S_2+n^2S_3)/(n(n-1)(n-2))
(6)
k_4=(-6S_1^4+12nS_1^2S_2-3n(n-1)S_2^2-4n(n+1)S_1S_3+n^2(n+1)S_4)/(n(n-1)(n-2)(n-3))
(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 n, the first few k-statistics are given by

k_1=mu
(8)
k_2=n/(n-1)m_2
(9)
k_3=(n^2)/((n-1)(n-2))m_3
(10)
k_4=(n^2[(n+1)m_4-3(n-1)m_2^2])/((n-1)(n-2)(n-3)),
(11)

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

var(k_1)=(kappa_2)/n
(12)
var(k_2)=(kappa_4)/n+(2kappa_2^2)/(n-1)
(13)
var(k_3)=(kappa_6)/n+(9kappa_2kappa_4)/(n-1)+(9kappa_3^2)/(n-1)+(6nkappa_2^3)/((n-1)(n-2))
(14)
var(k_4)=(kappa_8)/n+(16kappa_2kappa_6)/(n-1)+(48kappa_3kappa_5)/(n-1)+(34kappa_4^2)/(n-1)+(72nkappa_2^2kappa_4)/((n-1)(n-2))+(144nkappa_2kappa_3^2)/((n-1)(n-2))+(24(n+1)nkappa_2^4)/((n-1)(n-2)(n-3)).
(15)

An unbiased estimator for var(k_2) is given by

 var(k_2)^^=(2k_2^2n+(n-1)k_4)/(n(n+1))
(16)

(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

 var(k_3)^^=(6k_2^3n(n-1))/((n-2)(n+1)(n+3))
(17)

(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

M_1=mu
(18)
M_2=(N-n)/(n(N-1))mu_2
(19)
G_1=(N-2n)/(N-2)sqrt((N-1)/(n(N-n)))gamma_1
(20)
G_2=((N-1)(N^2-6Nn+N+6n^2)gamma_2)/(n(N-2)(N-3)(N-n))-(6N(Nn+N-n^2-1))/(n(N-2)(N-3)(N-n))
(21)

(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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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 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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k-Statistic

Cite this as:

Weisstein, Eric W. "k-Statistic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/k-Statistic.html

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