The problem of devising systematic policies for replacement of equipment subject to wear-out involves the detection of increases in failure rates. Detection procedures are defined as stopping timesNwith respect to the observed sequence of random failures. The concepts of “quickness of detection” and “frequency of false reactions” are made precise and a class of procedures is studied which optimizes the former asymptotically as the latter is reduced to zero. Results of Monte Carlo experiments are given which show that efficient quickness of detection is attainable simultaneously for various levels of increase in failure rates.