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k-statistics were introduced in a fundamental paper by R. A. Fisher (1928). The statistic of order p, written IC,, is defined as being the symmetric function of the independent variate values x,, x2, . . . , xm, whose mean value over all possible samples is K ~, the pth semi-invariant (or, in Fisher's terminology, cumulant) of the parent distribution. The value of the k-statistics rests chiefly on the relative simplicity which their use imports into certain branches of the theory of sampling. In this note I shall not be concerned with this aspect, but with the demonstration of some mathematical properties which they possess.
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M. G. Kendall (1940) studied this question.
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