Moments.

To avoid the large matrices of Sec. 7, consider just the sixteen ancestors of the eight configurations of three cells, for which in principle a matrix is required. There are only four nonzero frequencies, shown in the following variant of the previous table:

which gives rise to the following moments (the residuals remain when the powers of the dominant frequency are subtracted; the weights, to be used in Eq. 22, were approximated from Abramowitz and Stegun [34]):

Note how the residuals are eventually dominated by the second largest datum; of course the phenomenon would repeat if an attempt were made to separate more and more of the largest data from the rest. Even in this restricted example, at least ten moments are required to approximate to the dubious accuracy of , so the procedure may not be very practical. Nevertheless it is quite possible in theory, serving as a criterion whether vanishes or not, and suggesting that such vanishing might be unlikely unless the variance were exactly zero.

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Harold V. McIntosh
E-mail:mcintosh@servidor.unam.mx