The given engine has a reliability structure of four independent component-sockets connected in series, and canonically, a repair consists of complete part replacement in a failed component-socket. These two features let one model the number of system failures/repairs in a given time interval by a superimposed renewal process. The inter-failure times in each component-socket have been justified to have independent (using 2 and Kendall’s  test) and identical Weibull distributions (by computing bootstrap p-values of one and two sample Kolmogorov-Smironov tests based on the Kaplan-Meier estimate). Based on this model, both frequentist and Bayesian methodologies have been developed for point and interval estimation of the number of failures in a given time interval, the conditional and unconditional failure rates and two other newly introduced repairable system reliability metrics.

Next the information on the repair duration has been incorporated by modeling the parameters of the Weibull inter-failure times in each component-socket as functions of the preceding repair time. The nature of this functional relationship is being determined empirically en route to validating the assumption of Weibull inter-failure times (as mentioned above), by stratifying the repair times to fall in a few discrete strata and then analyzing the following inter-failure times. However this now distorts the renewal process structure, as the inter failure times in each component-socket are no longer identically distributed, and thus the expected number of failures in each component-socket cannot be computed using the renewal equation as before. The resulting process is being called a quasi-renewal process and the expected number of failures in a given time interval according to this process is being computed using a newly developed simulation technique. Using observations on the inter-failure times and the failed component-sockets, as before, this time together with repair times, both frequentist and Bayesian (MCMC based) methodologies have been developed for point and interval estimation of the five repairable system reliability metrics, mentioned in the last sentence of the last paragraph above.

In this problem however there are issues of appropriate model selection in the sense of the possibility of some model parameters being “insignificant”. In the frequentist frame work, this issue is settled by considering a series of nested models and then testing one model against the other using both Wald’s and likelihood ratio tests. In the Bayesian version of the problem, all possible models under a largest possible model, which need not be nested into one another, are considered and then the suitable one is chosen (in each component-socket) according to Berger and Perichi’s (Journal of American Statistical Association, 1997) Intrinsic Bayes Factor (IBF). The IBF computation requires a model jumping MCMC and for this purpose, the methodology suggested by Carlin and Chib (Journal of Royal Statistical Society, Series B, 1997) involving pseudo-priors has been used. In the final part of the thesis, a model proposed by Lawless and Thiagarjah (Technometrics, 1997), which models the conditional intensity function of the failure process having both (non-homogeneous Poisson process type) minimal and renewal (as has been considered in the rest of the thesis) repair characteristics, has been first extended to include repair times. Lawless and Thiagarjah suggested taking the nature of the minimal and renewal terms as linear or logarithmic functions of time. If the same is applied to the repair time term as well, then that gives rise to eight non-nested models. For the given data set, an appropriate model for each component-socket is chosen through IBF calculation as explained above.