Estimation of the lifetime distribution of the parts from the autopsy statistics of the machine

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Abstract
Given a coherent reliability system, let Z be the age of the machine at breakdown, and I the set of parts dead by time Z. We prove that if all lifetime distributions are non-atomic and share the same essential extrema, and if the incidence matrix of the minimal cut sets has rank equal to the number of parts, then the joint distribution of Z and I determines uniquely the lifetime distribution of each part. We present a Newton–Kantorovič iterative method for the computation of those distributions. We deal informally with the relaxation of the assumptions and with the statistical problem where instead of the joint distribution of Z and I we have an empirical estimate of this joint distribution.

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