How does our n-component system perform?

The statistical (black box) remaining lifetime of some simple coherent systems is stochastically compared with the remaining lifetime based on the information at hand. Parallel and cold standby systems of s-identical components with exponentially distributed lifetimes are considered. Information-based remaining lifetime is defined via the number of the nonfailed components at the moment of observation. The main question to answer is: Given the information on the current state of the system, is the remaining lifetime better (worse) in some stochastic sense than the remaining lifetime that is defined by the black box remaining lifetime? The result of this comparison for the general case depends on the time of observation and the time since observation. Several simple examples have been considered, because it is rather difficult technically to treat the stated problem in a greater generality. The main common feature of these examples, describing systems of s-identical components is the monotonically increasing system hazard-rate. An example with a nonmonotone hazard rate for a system of two different components has also been studied