Abstract: Suppose that identical systems are tested until failure and that each system is based on components whose lifetimes are independently and identically distributed with common continuous distribution function and survival function . Under the assumption that the system design isknown, Bhattacharya and Samaniego(2010) obtained the nonparametric maximum likelihood estimate of based on the observed system failure times and characterized its asymptotic behavior. The estimator studied in that paper has the form where is the system's reliability polynomial (see Barlow and Proshan (1981)) and is the empirical survival function of the system lifetimes . To treat this estimation problem when the system design is unknown, the design must be estimated from data. In this paper, we assume that auxiliary data in the form of a variable , the number of failed components at the time of system failure, is available along with the system's lifetime. Such data is typically available from a subsequent autopsy. The problem considered here is motivated by the fact that component reliability under field conditions is often not easily estimated through controlled laboratory tests. The data permits the estimation of the reliability polynomial (through the use of "system signatures"� -Samaniego (2007)). Denoting the estimated polynomial as , we study the properties of the estimator . Our main results include (1) is a -onsistent estimator of the component reliability function , (2) the asymptotic distribution of is normal and its asymptotic variance is given in closed form, and (3) the asymptotic variance of , based on the augmented data , is uniformly no greater than the asymptotic variance of , based on the data and the assumption that is known. This latter, perhaps surprising, result is confirmed in a variety of simulations and is illuminated through further heuristic onsiderations and further analysis.

Key words and phrases: Asymptotic efƒiciency, asymptotic normality, coherent system, component and system reliability, consistency, nonparametric estimation, NPMLE, nuisance parameter, system signature.