Preservation results for life distributions based on comparisons with asymptotic remaining life under replacements (Q2725296)

Let \(X>0\) be a random variable with finite mean, and let \(X_0\) be a r.v. distributed according to the asymptotic remaining life time of \(X\). The authors introduce two families of life time distributions: the class NBRUE (new better than renewal used in expectation) characterized by \(EX\geq E(X_0-t\mid X_0>t)\) for all \(t>0\) and the class HNBRUE (H for harmonic) defined by the property \(EX\geq (\int_0^t (E(X_0-x\mid X_0>x)^{-1} dx)/t)^{-1}\) for all \(t>0\). They show (supported by examples) the strict inclusions NBUE \(\subset\) NBRUE \(\subset\) HNBRUE and justify the introduction of the two classes by an economic interpretation. The main results of the paper focus on preservation properties with respect to convolutions and mixing.