AbstractOptimal inspection and replacement policies for a device subject to non-Markovian stochastic failures are derived in the case of inspections that may fail to provide a correct description of the state of the system. Two types of policies are considered, differing in the strategy followed in case of reported failure: one-test policies prescribe replacement, whereas two-test policies prescribe a more accurate work-up, designed to avoid the replacement of a functioning unit. For both cases and arbitrary failure distributions, general optimality conditions and algorithms that reduce the infinite horizon optimization to two dimensions are derived. Properties of the optimal policy, like monotonicity of the times between inspections, are studied. Methods for the implementation of fast computational algorithms are suggested.