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Polykay

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The symmetric statistic k_(r,s,...) defined such that

<k_(r,s,...)>=kappa_rkappa_s...,
(1)

where kappa_r is a cumulant. These statistics generalize k-statistic and were originally called "generalized k-statistics" (Dressel 1940). The term "polykay" was introduced by Tukey (1956; Rose and Smith 2002, p. 255). Polykays are commonly defined in terms of power sums, for example

k_(1,1)=(S_1^2-S_2)/((n-1)n)
(2)
k_(1,2)=(-S_1^3+(n+1)S_1S_2-nS_3)/((n-2)(n-1)n).
(3)

Polykays can be computed using PolyK[{r, s, ...}] in the Mathematica application package mathStatica.

See also

k-Statistic, Polyache

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References

Cook, M. B. "Bi-Variate k-Statistics and Cumulants of Their Joint Sampling Distribution." Biometrika 38, 179-195, 1951.Dressel, P. L. "Statistical Seminvariants and Their Setimates [sic] with Particular Emphasis on Their Relation to Algebraic Invariants." Ann. Math. Stat. 11, 33-57, 1940.Rose, C. and Smith, M. D. Mathematical Statistics with Mathematica. New York: Springer-Verlag, 2002.Tukey, J. W. "Keeping Moment-Like Computations Simple." Ann. Math. Stat. 27, 37-54, 1956.

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Polykay

Cite this as:

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

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