On k-order harmonic new better than used in expectation distributions
Let X be a nonnegative random variable with survival function defined by \(\bar F(\)x)\(\equiv P\{X>x\}\) and a finite mean \(\mu\). For \(k\geq 1\), the random variable X (or its survival function \(\bar F)\) is said to be k- order harmonic new better than used in expectation (k-HNBUE) if \(\mu^{- k}\leq t^{-1}\int^{t}_{0}\mu_ F^{-k}(x)dx\) for all \(t\geq 0\) where \(\mu_ F(x)\equiv (\int^{\infty}_{0}\bar F(y)dy)/\bar F(x).\) Similarly one defines X or \(\bar F\) to be k-order harmonic new worse than used in expectation (k-HNWUE). The authors study the classes of the k-HNBUE and k-HNWUE distributions. They find the relations between k-HNBUE and \((k+1)\)-HNBUE and between k- HNWUE and \((k+1)\)-HNWUE distributions. They demonstrate some counterexamples for possible conjectures. They obtain some closure properties of the k-HNBUE and k-HNWUE classes of distributions. Finally, they give bounds on the survival functions and on the moments of k-HNBUE and k-HNWUE random variables.
k-order harmonic new better than used in expectation
scientific article; zbMATH DE number 3888762