Estimating component characteristics from system failure‐time data

Suppose that failure times are available from a random sample of N systems of a given, fixed design with components which have i.i.d. lifetimes distributed according to a common distribution F. The inverse problem of estimating F from data on observed system lifetimes is considered. Using the known relationship between the system and component lifetime distributions via signature and domination theory, the nonparametric maximum likelihood estimator$ \hat{ \bar{F}} $N(t) of the component survival function $ \bar{F} $(t) is identified and shown to be accessible numerically in any application of interest. The asymptotic distribution of $ \hat{ \bar{F}} $N(t) is also identified, facilitating the construction of approximate confidence intervals for $ \bar{F} $(t) for N sufficiently large. Simulation results for samples of size N = 50 and N = 100 for a collection of five parametric lifetime models demonstrate the utility of the recommended estimator. Possible extensions beyond the i.i.d. framework are discussed in the concluding section.