Reliability interpretations of some concepts from economics

In a recent article, Chandra and Singpurwalla have pointed out the close relationship between the Lorenz curve, which is frequently used when illustrating income distributions in economics, and the total-time-on-test transform (TTT transform), which has proved to be a very useful tool in reliability. They also presented some characterizations of aging properties by using the Lorenz curve. The purpose of this article is to point out some further results in the same area and to give reliability interpretations of some common measures of income inequality.     

The hnbue and hnwue classes of life distributions

Different properties of the HNBUE (HNWUE) class of life distributions (i.e.), for which \documentclass{article}\pagestyle{empty}\begin{document}$\int_t^\infty {\,\,\,\mathop F\limits^-(x)\,dx\, \le \,(\ge)\,\mu }\]$\end{document} exp(?t/μ) for t ≥ 0, where μ = \documentclass{article}\pagestyle{empty}\begin{document}$\int_t^\infty {\,\,\,\mathop F\limits^-(x)\,dx}$\end{document} are presented. For instance we characterize the HNBUE (HNWUE) property by using the Laplace transform and present some bounds on the survival function of a HNBUE (HNWUE) life distribution. We also examine whether the HNBUE (HNWUE) property is preserved under the reliability operations (i) formation of coherent structure, (ii) convolution and (iii) mixture. The class of distributions with the discrete HNBUE (discrete HNWUE) property (i.e.), for which \documentclass{article}\pagestyle{empty}\begin{document}$\sum\limits_{j=k}^\infty {\mathop{\mathop P\limits^-_{j\,\,\,}\, \le(\ge)\,\mu(1 - 1/\mu)^{^k }}\limits^{}} $\end{document} for k = 0, 1, 2, ?, where μ =\documentclass{article}\pagestyle{empty}\begin{document}$\sum\limits_{j=0}^\infty {\mathop {\mathop P\limits^- _{j\,\,\,\,\,}and\mathop P\limits^ - _{j\,\,\,\,\,}=}\limits^{}}\,\,\sum\limits_{k=j+1}^\infty {P_k)}$\end{document} is also studied.     

TTT transforms and age replacements with discounted costs

The ordinary age replacement problem consists of finding an optimal age at which a unit, needed in a continuous production process, should be replaced in order to minimize the average long-run cost per unit time. Bergman introduced a graphical procedure based on the total-time-on-test (TTT) concept for the analysis of the age replacement problem. In this article, that idea is generalized to the situation of discounted costs. We also study a more general age replacement problem in which we have a form of imperfect repair.