Polykay

The symmetric statistic  defined such that

(1)

where  is a cumulant. These statistics generalize k-statistic and were originally called "generalized -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

			(2)
			(3)

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

REFERENCES:

Cook, M. B. "Bi-Variate -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.