Abstract
Suppose that N identical systems are tested until failure and that each system is based on n components whose lifetimes are independently and identically distributed with common continuous distribution function F(t) and survival function (F) over bar (t) = 1- F(t). Under the assumption that the system design is known, Bhattacharya and Samaniego (2010) obtained the nonparametric maximum likelihood estimate of F based on the observed system failure times and characterized its asymptotic behavior. The estimator studied in that paper has the form (sic)(o) (t) = h(-1) [(sic)(T)(t)] where h(.) is the system's reliability polynomial (see Barlow and Proshan (1981)) and (sic)(T)(t) is the empirical survival function of the system lifetimes {T-1,...,T-N}. 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 K, 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 (T-1, K-1), (T-2, K-2),..., (T-N, K-N) permits the estimation of the reliability polynomial h (through the use of "system signatures" - Samaniego (2007)). Denoting the estimated polynomial as (h) over cap, we study the properties of the estimator (sic)(t) = (h) over cap (-1) [(sic)(T)(t)]. Our main results include (1) (sic)(t) is a root n-consistent estimator of the component reliability function (F) over tilde (t), (2) the asymptotic distribution of (F) over bar (t) is normal and its asymptotic variance is given in closed form, and (3) the asymptotic variance of (sic)(o)(t), based on the augmented data {T-i, K-z}, is uniformly no greater than the asymptotic variance of (sic)(o)(t), based on the data {T-i} and the assumption that h is known. This latter, perhaps surprising, result is confirmed in a variety of simulations and is illuminated through further heuristic considerations and further analysis.