Decision Making with Imprecise Probabilities

Nonparametric predictive inference for system reliability using the survival signature

“…. , l K , 0) follow by Equation (9), and together these fully specifyΦ. While this does require new computations, the overall system structure remains the same, and attention can now be restricted to l k − 1 components of type k, with the replaced component assumed to be functioning.…”

“…to study effects of additional assumptions underlying other statistical methods. NPI uses lower and upper probabilities, also known as imprecise probabilities, to quantify uncertainty [9,19,39,40] and has strong consistency properties from frequentist statistics perspective [8,13]. NPI provides a solution to some explicit goals formulated for objective (Bayesian) inference, which cannot be obtained when using precise probabilities [12], and it never leads to results that are in conflict with inferences based on empirical probabilities.…”

“…This is easier than the original calculation of Φ in Table 1, because it concerns the same system structure but with the new component of type 3 in Figure 3 certainly functioning. Given the fully specified survival signature Φ of the original system and the values ofΦ(l 1 ,l 2 , 1), the values ofΦ(l 1 ,l 2 , 0) in Table 5 are easily calculated using Equation (9). …”

On the structure function and survival signature for system reliability

“…quite straightforward application of Bayesian statistical methods, where F k (t) may be assumed to belong to a parametric family but where also nonparametric approaches are possible, both are illustrated by Aslett et al [2]. Coolen et al [10] illustrate nonparametric predictive inference [3,6] for the system survival function using equation (2), where the ciid assumption is not made and with the generalization to imprecise probabilities [4]. Feng et al [12] illustrate the use of the survival signature combined with known sets of probability distributions for the failure times of the components of different types, so also within theory of imprecise probability [4], and they also discuss computation of importance measures for components in the systems using the survival signature.…”

“…Coolen et al [10] illustrate nonparametric predictive inference [3,6] for the system survival function using equation (2), where the ciid assumption is not made and with the generalization to imprecise probabilities [4]. Feng et al [12] illustrate the use of the survival signature combined with known sets of probability distributions for the failure times of the components of different types, so also within theory of imprecise probability [4], and they also discuss computation of importance measures for components in the systems using the survival signature. To get further insight into the system survival function for complex systems, when analytical derivations are not feasible anymore, the ability to perform simulations efficiently is crucial.…”