Exact Lower Moments of Order Statistics in Samples from the Chi- Distribution (1 d.f.)

Abstract

Numerous contributions have been made to the problem of order statistics in samples from normal and exponential populations. For the problem of location with symmetry Fraser [1] derived a locally most powerful rank test against normal alternatives. It is the Wilcoxon test statistic with the ranks replaced by the corresponding expected values of order statistics in a sample from the chi-distribution with one degree of freedom. Gupta [5] considered the order statistics from the standardized gamma distribution with the parameter $r$ defined on the positive integers (that is, from the chi-distribution with even degrees of freedom) and derived expressions for the $k$th moments of an order statistic and the covariance between two order statistics. He also presented a table of numerical values of the $k$th moments of an order statistic accurate to six magnificant digits for $k = 1 (1) N, N \leqq 15$ and $r = 1 (1) 5$, where $N$ is the sample size. It might be of interest to consider the problem of order statistics in samples from chi-populations with odd degrees of freedom. However, this problem seems to be more difficult than the one considered by Gupta [5]. In the present paper, the expected values for samples to size four and the mixed and second moments (about the origin) for samples to size five, drawn from the chi-population (1 d.f.) have been evaluated. Numerical values of these to eight decimal places are computed. Section 2 contains general formulae and some definite integrals used in the computation. The results in Section 3 have theoretical interest in showing the relationships between moments of order statistics from chi (1 d.f.) and the standard normal distributions. In Section 4, there is a discussion about the number of integrals required to evaluate the first, second and mixed moments of order statistics for each $N$, given these moments to $N - 1$ and the existence of the tables for the normal distribution. There is also a discussion about the cumulative rounding error involved in using the formulae recurrently.