Simple bounds for counting processes with monotone rate of occurrence of failures

The article discusses some aspects of analogy between certain classes of distributions used as models for time to failure of nonrepairable objects, and the counting processes used as models for failure process for repairable objects. The notion of quantiles for the counting processes with strictly increasing cumulative intensity function is introduced. The classes of counting processes with increasing (decreasing) rate of occurrence of failures are considered. For these classes, the useful nonparametric bounds for cumulative intensity function based on one known quantile are obtained. These bounds, which can be used for repairable objects, are similar to the bounds introduced by Barlow and Marshall [Barlow, R. Marshall, A. Bounds for distributions with monotone hazard rate, I and II. Ann Math Stat 1964; 35: 1234–74] for IFRA (DFRA) time to failure distributions applicable to nonrepairable objects.