ing failure rate average” [IFRA].) Another model of aging is “new is better than used” (NBU). For systems with time-to-failure distributions having this characteristic, the probability of lasting t units of time when the system is new is greater than the probability of lasting an additional t units of time given that one such system has already lasted ∆ units. (This notion is slightly distinct from IFR since it links the performance of older systems to that of a new system and not to the performance of intermediate aged systems.) Another widely used modeling assumption is that of decreasing mean residual life (DMRL). This assumption characterizes a time-to-failure distribution in which the expected additional or residual lifetime of a system of age t is a decreasing function of t. This concept is distinguished from IFR since it relates mean lifetimes rather than lifetime probabilities.

Samaniego argued that instead of assuming a specific distributional form for the time-to-failure distribution and estimating parameters to identify a particular member of these distributional families, one could estimate the lifetime distribution under one of the above nonparametric assumptions. For example, under the assumption that the time-to-failure distribution is IFR, the nonparametric maximum-likelihood estimate of the hazard rate (the instantaneous failure rate conditional on the event that the system has lasted until time t, which is essentially equivalent to estimation of the time-to-failure distribution) at time t is a nondecreasing step function whose computation involves the well-understood framework of isotonic regression. Similar constraints from assumptions such as NBU produce alternative nonparametric estimators.

These are one-sample techniques for the problem of estimating the properties of a single time-to-failure distribution. With respect to the problem of combining information, the natural situation is that of comparing samples from two related experiments. Rather than make the linked-parameter assumption of Samaniego et al. (2001) (i.e., the λ factor), Samaniego instead used nonparametric assumptions about the relationship between the time-to-failure distributions for developmental and operational testing of a system. Three well-known formulations of the notion that a sampled quantity (failure time) from one distribution tends to be smaller than a sampled quantity from another distribution are as follows (see Shaked and Shanthikumar [1994] for further details): (1) stochastic ordering, when the probability that the next failure will be t time units or greater for a system in developmental test is greater than the probability that the next failure will be t time units or greater for the same system in operational test, for all t; (2) hazard-rate ordering, when the instantaneous failure rate,



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