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Some characterizations of renewal densities with emphasis in reliability. (English) Zbl 0563.62077
Let a component operating in a system be replaced upon failure by another component having the same lifetime distribution F so that the sequence of component life lengths forms a renewal process. Let $$U_ t$$ be the age of the component operating at time t and let $$V_ t$$ be the remaining life of the component. The author shows the following:
(i) If F is NBUE (NWUE) and the mean life is equal to the mean residual life, then the renewal distribution is exponential.
(ii) If the renewal distribution belongs to the one parameter linear exponential family and the mean life is equal to the mean residual life, then F is exponential.
(iii) If the renewal distribution belongs to the one parameter linear exponential family and the total mean life is twice the mean residual life, then F is exponential.
(iv) If F has IFR (DFR), then Cov(U,V)$$\leq (\geq)0$$, but the converse is not true.