We focus on obtaining Block–Savits type characterizations for different ageing classes as well as some important renewal classes by using the Laplace transform. We also introduce a novel approach, based on the equilibrium distribution, to handle situations where the techniques of Block and Savits (1980) either fail or involve tedious calculations. Our approach in conjunction with the theory of total positivity yields Vinogradov’s (1973) result for the increasing failure rate class when the distribution function is continuous. We also present simple but elegant proofs for Block and Savits’ results for the decreasing mean residual life, new better than used in expectation, and harmonic new better than used in expectation classes as applications of our approach. We address several other related issues that are germane to our problem. Finally, we conclude with a short discussion on the issue of convolutions.

Suppose that a device is subjected to shocks governed by a counting process N = {N(t), t ≧0}. The probability that the device survives beyond time t is then H̄(t)=Σk=0 ∞Q̄ℙ[N(t)=k], where Q̄k is the probability of surviving k shocks. It is known that H is NBU if the interarrivals Uk , ∊ ℕ+, are independent and NBU, and Q̄ k+j ≦ Q̄k · Q̄j holds whenever k, j ∊ ℕ. Similar results hold for the classes of the NBUE and HNBUE distributions. We show that some other ageing classes have similar properties.