On the Moments of Order Statistics in Samples From Normal Populations

The purpose of this paper is to show the geometrical significance of the maments of order statistic derived from normal populations. It appears that these, as well as the moment-generating function of the sqare of any order statistic, are intimately related to the contents of the members of a class of hyperspherical simplices. Further geometrical interpretations, together with the extension of the present results to bivariate moments and other properties, will be provided in subsequent publications. The problem of order statistics in normal populations has been extensively considered in the literature. For example,† Tippett (1925) gives the second, third and fourth moments of the extreme order statistics for a few sample sizes. Hojo (1931) examinses the sampling variation of the median, quartiles and interquartile distance in samples from normal populations and computes a number of integrals for this purpose. Cole (1951) produces a very simple recurrence relationship between the ‘normalized’ moments which enables the normalized moments for all samples of size no greater than n to be obtained, by successive differcing from the ‘normalized’ moments of the extreme order statistics in samples of size m≦n.‡ Hastings. Mosteller, Tukey & Winsor (1947) give, among other results, the means, variances, covariances and correlations of order statistics in samples of ten or less from a normal poputation, some of the results being given to only two decimal places because of the extreme labour of the computation. Jones (1948) and Godwin (1949a, b) have obtained exact values for same of the lower moments. In particular, Godwin (1949a, p. 283) proceeds in a more systematic manner, and approaches most closely the essential idea upon which this paper is based. In general, it may be said that many statisticians have attacked the problem in a rather disjointed and fragmentary manner, usually from the small sample end, but have failed to devalop a systametic attack which shall at the same time throw light on the interconnexion between the moments and enable the computation of the moments to become an economic proposition. It is believed that these conditions are met by this paper.