On classes of lifetime distributions with unknown age (Q2711565)

\textit{A.A. Alzaid} [Commun. Stat., Stochastic Models 10, No. 3, 649--659 (1994; Zbl 0817.60095)] introduced the used better than aged (in expectation), UBA (UBAE), class of distributions of a nonnegative lifetime \(X\). In the present paper, after a little re-working of the definitions, the relationships between UBA, UBAE and established classes, e.g., DMRL, DVRL, are discussed. Integer \(X\) are then considered, with the associated D-UBA class being defined by \(1< z_0< \infty\) and \(P(X> n+1\mid X>n)\geq {1/z_0}\), \(n= 0,1,\dots\), where \(z_0\) is the radius of convergence of the PGF of \(X\). D-UBA is shown to contain D-DMRL, and to admit a characterization in terms of Poisson-UBA mixtures. Finally, the distribution transform NEWLINE\[NEWLINEF(\cdot)\to 1- \int^\infty_0 e^{-\alpha t}(1- F(\cdot+ t)) dt\biggl/\int^\infty_0 e^{-\alpha t}(1- F(t)) dt,\quad -\infty< \alpha<\infty,NEWLINE\]NEWLINE which arises in queueing and ruin, is shown to preserve UBA and UBAE properties.