SOME EXACT EXPRESSIONS FOR THE MEAN AND HIGHER MOMENTS OF FUNCTIONS OF SAMPLE MOMENTS

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

1Mathematical Sciences Section, Engineering Physics and Mathematics Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6367, U.S.A.

2Computer 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 → ∞, 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.