On Moments of Order Statistics and Quasi-ranges from Normal Populations

Abstract

The main purpose of the paper is to obtain the lower bound on the number of integrals to be evaluated in order to know the first, second and mixed (linear) moments of the normal order statistics ().S.) in a sample of size $N$ assuming that these moments are available for sample sizes less than $N$. Towards this, the recurrence relationships, identities, etc. among the moments of the normal order statistics, which have appeared in the literature have been collected with appropriate references. Also, these formulae are listed and stated in the most general form wherever possible. Simple and alternate proofs of some of these formulae are given. These results are also supplemented with new formulae or relationships. It is shown that it is sufficient to evaluate at most one single integral and ($N$-2)/2$ double integrals when $N$ is even and one single integral and ($N$-3)/2$ double integrals when $N$ is odd, in order to know the first, second and mixed (linear) moments of normal O.S. However, for these moments of O.S. in samples drawn from an arbitrary population symmetric about zero, one has to evaluate one more double integral in addition to the number of integrals required in the case of normal O.S. Also, a possible scheme of computing these moments which will be useful especially for small sample sizes, is presented in Section 5. The lower moments of quasi-ranges in samples drawn from an arbitrary population symmetric about zero are expressed in terms of the moments of the0 corresponding O.S. Simple recurrence formulae among the expected values of quasi-ranges in samples drawn from an arbitrary continuous population are obtained. A modest list of references is provided at the end which is by no means exhaustive.