Some notes on unobserved parameters (frailties) in reliability modeling

• We discuss and clarify the notion of frailty in reliability context and present and analyze several meaningful examples.
• The paper provides a new insight and general perspective on reliability models with unobserved parameters.
• The main message of the paper is well illustrated by several meaningful examples and emphasized by detailed discussion.

Unobserved random quantities (frailties) often appear in various reliability problems especially when dealing with the failure rates of items from heterogeneous populations. As the failure rate is a conditional characteristic, the distributions of these random quantities, similar to Bayesian approaches, are updated in accordance with the corresponding survival information. At some instances, apart from a statistical meaning, frailties can have also useful interpretations describing the underlying lifetime model. We discuss and clarify these issues in reliability context and present and analyze several meaningful examples. We consider the proportional hazards model with a random factor; the stress–strength model, where the unobserved strength of a system can be viewed as frailty; a parallel system with a random number of components and, finally, the first passage time problem for the Wiener process with random parameters.