The mean time to failure (MTTF) function in age replacement is used to evaluate the performance and effectiveness of the age replacement policy. In this paper, based on the MTTF function, we introduce two new nonparametric classes of lifetime distributions with nonmonotonic mean time to failure in age replacement; increasing then decreasing MTTF (IDMTTF) and decreasing then increasing MTTF (DIMTTF). The implications between these classes of distributions and some existing classes of nonmonotonic ageing classes are studied. The characterizations of IDMTTF and DIMTTF in terms of the scaled total time on test transform are also obtained.

a b s t r a c tThe purpose of this paper is to study the component importance in k-out-of-n system with components having random weights. The system works if at least k components work and the total weight of all working components is above the required level c. A new measure of component importance, called weighted importance, for such a system is introduced and its relation with Birnbaum reliability measure of importance is studied. It is shown that the larger weight stochastically implies the larger weighted measure of importance. It is also shown that if two components are identically distributed then the larger weight stochastically implies the larger Birnbaum reliability measure of importance. Finally, it is proved that if the weight of a component is stochastically increased, then the weighted and Birnbaum reliability measures of importance are increased accordingly.

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In this paper, we study the problem of optimal allocation of insurance layers for a portfolio of i.i.d exponential risks. Using the first stochastic dominance criterion, we obtain an optimal allocation for the total retain risks faced by a policyholder. This result partially generalizes the known result in the literature for deductible as well as policy limit coverages.