The Annals of Mathematical Statistics

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

Zakkula Govindarajulu

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  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  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 . 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.

Article information

Source
Ann. Math. Statist., Volume 33, Number 4 (1962), 1292-1305.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177704362

Digital Object Identifier
doi:10.1214/aoms/1177704362

Mathematical Reviews number (MathSciNet)
MR141179

Zentralblatt MATH identifier
0218.62045

JSTOR