MOMENT COEFFICIENTS OF THE k-STATISTICS IN SAMPLES FROM A FINITE POPULATION

Tukey (1950) has shown how it is possible to simplify the presentation of the moment coefficients of the distribution of certain of the k-statistics in samples from a finite population, illustrating in particular by calculating the variances of k, and k2, and their covariance. His method is to introduce a new sample statistic krs. , and to work out once and for all certain non-linear functions of these as linear functions of themselves. This is much of the labour of calculating the sampling moment coefficients; the subsequent work consists of selecting those that are required and putting them together. It should be mentioned that Dressel (1940), following the work of Dwyer (1938), introduced statistics L (rs, 1) as unbiased estimates of products ArAs... of the Thiele seminvariants. These are the same quantities as Tukey's kr8 although, because of the stated limitation to the values of r, 8, ..., they do not form a complete set. It is of some interest to carry the calculations further and derive new results. In this paper the formulae are extended to the 6th order, and all moment coefficients evaluated to this order. This will carry us a good way beyond the results of Irwin & Kendall (1944). Finally, certain basic formulae are given for the 7th and 8th orders. 2. Using the notation of David & Kendall (1949) for the symmetric functions of the n sample observations, we may write krs in general as