Kendall Operator

The operator that can be used to derive multivariate formulas for moments and cumulants from corresponding univariate formulas.

For example, to derive the expression for the multivariate central moments in terms of multivariate cumulants, begin with

(1)

Now rewrite each variable as to obtain

(2)

Now differentiate each side with respect to , where

(3)

and wherever there is a term with a derivative , remove the derivative and replace the argument with times itself, so

(4)

Now set any s appearing as coefficients to 1, so

(5)

Dividing through by 4 gives

(6)

Finally, set any coefficients powers of appearing as term coefficients to 1 and interpret the resulting terms as , so that the above gives

(7)

This procedure can be repeated up to times, where is the subscript of the univariate case.

Iterating the above procedure gives

			(8)
			(9)
			(10)
			(11)
			(12)

giving the identities

			(13)
			(14)
			(15)
			(16)
			(17)