Estimation for a probabilistic stress-strength model

Estimation for an unknown strength distribution is considered in two situations: (1) independent identically distributed stresses are applied to the component until it fails (no cumulative damage); and (2) each applied stress causes damage to the component and damage cumulates until the component fails. Both situations lead to mixtures of probability distributions, with the strength distribution playing the role of the mixing distribution. Based on the observation of cycles to failure of several independent components and on the theory of mixtures of distributions, estimators of the mixing distributions are obtained using linear programming. In particular, the solution to the linear-programming problem yields a probability mass function which approximates the unknown strength distribution. From this estimate of the strength distribution, an estimate of the mean strength of the item can be obtained by the usual computation of the mean of a probability distribution. Hence, the results provide a method of estimating the mean strength, or other parameters of the strength distribution, without requiring observations directly on the strengths of the test components