Nonparametric predictive inference for reliability of a k-out-of-m:G system with multiple component types Ahmad M. Aboalkhair a , Frank P.A. Coolen b,n , Iain M. MacPhee b,1 a Department of Applied Statistics and Insurance, Mansoura University, Mansoura 35516, Egypt b Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK article info Keywords: Diversity Lower and upper probabilities Nonparametric predictive inference Redundancy System reliability k-out-of-m:G system abstract Nonparametric predictive inference for system reliability has recently been presented, with specific focus on k-out-of-m:G systems. The reliability of systems is quantified by lower and upper probabilities of system functioning, given binary test results on components, taking uncertainty about component functioning and indeterminacy due to limited test information explicitly into account. Thus far, systems considered were series configurations of subsystems, with each subsystem i a k i -out-of-m i :G system which consisted of only one type of components. Key results are briefly summarized in this paper, and as an important generalization new results are presented for a single k-out-of-m:G system consisting of components of multiple types. The important aspects of redundancy and diversity for such systems are discussed. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Lower and upper probabilities generalize the standard theory of (‘single-valued’ or ‘precise’) probability and provide a powerful method for uncertainty quantification, see Utkin and Coolen [12] for an introductory overview from the perspective of reliability theory and applications and many further references. The main idea is that, for an event A, a lower probability P ðAÞ and upper probability P ðAÞ are specified, such that 0 r P ðAÞ r P ðAÞ r1, with classical precise probability appearing in the special case with P ðAÞ¼ P ðAÞ. Like precise probability, lower and upper probabilities have different possible interpretations, including a subjective interpretation in terms of buying prices for gambles. Informally, a lower probability P ðAÞ can be interpreted as reflecting the evidence in support of event A, which makes focus on lower probability for system functioning natural and attractive in relia- bility studies, we use this as the reliability measure of interest throughout this paper. For completeness, however, we also present the corresponding upper probability P ðAÞ, which can be inter- preted by considering that 1  P ðAÞ reflects the evidence against event A, so in support of the complementary event A c . The lower and upper probabilities presented in this paper are naturally linked by the conjugacy property P ðAÞ¼ 1  P ðA c Þ [1]. Zio [15] mentions the need for research into quantification of uncertainty in reliability by means of representations other than probability distributions. The lower and upper probabilities in NPI, as used in this paper, are optimal bounds on probabilities resulting from making only few assumptions. They enable statistical inference that can be considered ‘objective’ in a way that is not possible when restricting to classical probabilities [3]. Coolen [2] presented lower and upper predictive probabilities for Bernoulli random quantities, based on an assumed underlying latent variable model, with future outcomes of random quantities related to data by the assumption A ðnÞ introduced by Hill [6]. These lower and upper probabilities are part of a wider statistical methodology called ‘Nonparametric Predictive Inference’ (NPI), which is a frequentist statistical approach with strong consistency properties [1], see Coolen [3] for an overview and further refer- ences (see also www.npi-statistics.com). Coolen-Schrijner et al. [5] presented NPI for system reliability, in particular for series systems with subsystem i a k i -out-of-m i :G system. Such systems are common in practice, and can offer the important advantage of building in redundancy by increasing some m i to increase the system reliability. Coolen-Schrijner et al. [5] considered the situation where each subsystem consists of components of a single type, with different subsystems having different types of components. They applied NPI for Bernoulli data to such systems, with inferences on each subsystem i based on information from tests on n i components, where the tested components are assumed to be exchangeable with the corre- sponding components to be used in that subsystem. Only situa- tions where components and the system either function or not when called upon were considered, we make the same assumption Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ress Reliability Engineering and System Safety http://dx.doi.org/10.1016/j.ress.2014.04.010 0951-8320/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author. E-mail address: frank.coolen@durham.ac.uk (F.P.A. Coolen). 1 Iain passed away in January 2012. We dedicate this paper, to which he fully contributed, to his memory. Please cite this article as: Aboalkhair AM, et al. Nonparametric predictive inference for reliability of a k-out-of-m:G system with multiple component types. Reliability Engineering and System Safety (2014), http://dx.doi.org/10.1016/j.ress.2014.04.010i Reliability Engineering and System Safety ∎ (∎∎∎∎) ∎∎∎–∎∎∎