Nonparametric Bayes Estimation of Reliability of a Coherent System

Simultaneous estimation of system and components reliability is considered when independent partition-based Dirichlet(PBD) prior is assigned on components distribution. Denote the lifetime of component j in the i-th system by {Tij , j = 1, 2, 3, . . . ,K} and the end of monitoring time by {τi, i = 1, 2, . . . , n}. Assume that {Tij , i = 1, 2, 3, . . . , n} and {τi, i = 1, 2, . . . , n} are IID with distribution Fj and G, respectively and with {Tij}s and {τi}s mutually independent and Tij and Til also independent for j 6= l. In our nonparametric Bayesian approach we assign independent partition-based Dirichlet(PBD) prior, D(αj), on Fj , j = 1, 2, . . . ,K, with parameter αj , a non-null finite measure on <+. We derive the nonparametric Bayes estimator of component reliabilities, F̄j = 1 − Fj for j = 1, 2, . . . ,K, and an estimator of system reliability function is F̄φ(t) = hφ(F̄1(t), F̄1(t), . . . , F̄K(t)), where φ is the structure function of the system. The product-limit (PL) type estimator of system reliability presented in Doss et al. (Annals, 89) is a special case of our estimator, obtained by letting αj(<+) → 0 for j = 1, 2, . . . ,K. Through simulation studies, we demonstrate that the PL-type estimator has smaller bias but higher root-mean-squared errors (RMSE) than our proposed estimator. Even when the prior mean does not coincide with the true distribution function, Bayes estimator has smaller or equal RMSE than the PL-type estimator with smaller value of precision parameter, indicating robustness of our estimator. In addition, our proposed estimator is smoother than the PL-type estimator.