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SOME EXACT EXPRESSIONS FOR THE MEAN AND

HIGHER MOMENTS OF FUNCTIONS OF SAMPLE MOMENTS

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K. O. BOWMAN^{1} AND L. R. SHENTON^{2}

^{1} *Mathematical Sciences Section, Engineering Physics and Mathematics Division,*

Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6367, U.S.A.

^{2} *Computer Services Annex, University of Georgia, Athens, GA 30602, U.S.A.*
(Received October 8, 1990; revised February 5, 1992)

**Abstract.**
Examples of exact expressions for the moments (mainly
of the
mean) of functions of sample moments are given. These provide checks on
alternative
developments such as asymptotic series for *n* \to \infty, and simulation
processes.
Exact expressions are given for the mean of the square of the sample
coefficient of
variation, particularly in uniform sampling; Frullani integrals studied by G. H.
Hardy arise. It should be kept in mind that exact results for (joint) moment
generating functions (mgfs) are of interest as they produce a means of obtaining
exact results for (cross) moments--including moments with negative
indices. Thus an
exact expression for the joint mgf of the 1st two noncentral moments can be
used to
obtain the mean of the (*c*.*v*.)^{2} (but not for the mean of the *c*.*v*.). A
general
expression is given for the moment generating function of the sample
variance. The
limitations of Fisher's symbolic formula for the characteristic function of
sample
moments (or more general statistics) are noted.

*Key words and phrases*:
Coefficient of variation, Frullani
integrals,
moment series, sample variance, symbolic characteristic function.

**Source**
( TeX ,
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