Recurrence Relations and Identities for Order Statistics

Abstract

Order statistics and their moments have been of great interest from the turn of this century since Sir Francis Galton (1902) and Karl Pearson (1902) studied the distribution of the difference of two successive order statistics. The moments of order statistics did, subsequently, assume considerable importance in the statistical literature and have been numerically tabulated extensively for several distributions. For example, one can refer to Harter (1970a,b) and David (1981) for a detailed list of these tables. Meanwhile, with the primary intention of reducing the amount of direct computation of these moments, many authors including Jones (1948), Godwin (1949), Cole (1951) and Sillitto (1951, 1964) carried out independent investigations and derived several recurrence relations and identities satisfied by these moments of order statistics. Many of these relations and identities are quite useful as they express the higher order moments in terms of the lower order moments thus making the evaluation of higher order moments easy and, in addition, provide some simple checks to test the accuracy of the computation of moments of order statistics. It was only 25 years ago, however, that Govindarajulu (1963a) nicely summarized all these results and established some more recurrence relations and identities satisfied by the single and the product moments of order statistics. He then systematically applied these results in order to determine the maximum number of single and double integrals to be evaluated for the calculation of means, variances and covariances of order statistics in a sample of size n, assuming these quantities for all sample sizes less than n to be known. By a simple generalization of one of the results of Govindarajulu (1963a), Joshi (1971) determined that for distributions symmetric about zero the number of double integrals to be evaluated for even values of n is in fact zero. Recently, Joshi and Balakrishnan (1982) established similar results for any arbitrary continuous distribution and applied them to improve over the bounds of Govindarajulu. Yet another interesting application of these recurrence relations and identities among order statistics is in establishing some combinatorial identities and this has been demonstrated by Joshi (1973) and Joshi and Balakrishnan (1981a).