Polyache

The statistics  defined such that

where  is a central moment. These statistics generalize h-statistics and were originally called "generalized -statistics" (Tracy and Gupta 1974). The term "polyache" was introduced by Rose and Smith (2002, p. 255) by way of analogy with the polykay statistic. Polyaches are commonly defined in terms of power sums, for example

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

REFERENCES:

Rose, C. and Smith, M. D. Mathematical Statistics with Mathematica. New York: Springer-Verlag, pp. 253-256, 2002.

Tracy, D. S. and Gupta, B. C. "Generalized -Statistics and Other Symmetric Functions." Ann. Stat. 2, 837-844, 1974.