Estimation of the Guarantee Time in an Exponential Failure Model

The location parameter μ in a 2-parameter exponential failure model is often termed the guarantee time since no failure can occur until the life exceeds μ. Several point estimators of μ are compared by examining each estimator's risk corresponding to a loss function which features a greater penalty for overestimating than for underestimating μ. The estimators considered in this fashion are: maximum likelihood, minimum variance unbiased, best invariant, and median unbiased. A new estimator of μ is derived which has smaller risk under our assumptions than any of the aforementioned estimators. A 2-fold pairwise comparison technique is then used which examines a) the odds in favor of the new estimator's being closer to μ than is a competitor, and b) each estimator's risk not only when it is closer to μ than is the other but also when it is not.