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September, 1985 A Measure of Variability Based on the Harmonic Mean, and Its Use in Approximations
Ann. Statist. 13(3): 1239-1243 (September, 1985). DOI: 10.1214/aos/1176349668
Let $X$ be a positive random variable and assume that both $a = EX^{-1}$ and $\mu = EX$ are finite. Define $c^2 = 1 - (a\mu)^{-1}$. This quantity serves as a measure of variability for $X$ which is reflected in the behavior of completely monotone functions of $X$. For $g$ completely monotone with $g(0) < \infty$: $0 \leq Eg(X) - g(EX) \leq c^2g(0) \text{and}\operatorname{Var} g(X) \leq c^2g^2(0).$
Mark Brown. "A Measure of Variability Based on the Harmonic Mean, and Its Use in Approximations." Ann. Statist. 13 (3) 1239 - 1243, September, 1985. https://doi.org/10.1214/aos/1176349668
Published: September, 1985
First available in Project Euclid: 12 April 2007
zbMATH: 0572.62026
MathSciNet: MR803770
Digital Object Identifier: 10.1214/aos/1176349668
Secondary: 62N99
Keywords: Completely monotone functions, harmonic mean, Laplace transforms, NWUE distributions, variability
Rights: Copyright © 1985 Institute of Mathematical Statistics
Ann. Statist.
Vol.13 • No. 3 • September, 1985
Mark Brown "A Measure of Variability Based on the Harmonic Mean, and Its Use in Approximations," The Annals of Statistics, Ann. Statist. 13(3), 1239-1243, (September, 1985)