Finite Mixture Models: A Key Tool for Reliability Analyses

Marko Nagode, Simon Oman, Jernej Klemenc and Branislav Panić ()
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Marko Nagode: Faculty of Mechanical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia
Simon Oman: Faculty of Mechanical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia
Jernej Klemenc: Faculty of Mechanical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia
Branislav Panić: Faculty of Mechanical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia

Mathematics, 2025, vol. 13, issue 10, 1-24

Abstract: As system complexity increases, accurately capturing true system reliability becomes increasingly challenging. Rather than relying on exact analytical solutions, it is often more practical to use approximations based on observed time-to-failure data. Finite mixture models provide a flexible framework for approximating arbitrary probability density functions and are well suited for reliability modelling. A critical factor in achieving accurate approximations is the choice of parameter estimation algorithm. The REBMIX&EM algorithm, implemented in the rebmix R package, generally performs well but struggles when components of the finite mixture model overlap. To address this issue, we revisit key steps of the REBMIX algorithm and propose improvements. With these improvements, we derive parameter estimators for finite mixture models based on three parametric families commonly applied in reliability analysis: lognormal, gamma, and Weibull. We conduct a comprehensive simulation study across four system configurations, using lognormal, gamma, and Weibull distributions with varying parameters as system component time-to-failure distributions. Performance is benchmarked against five widely used R packages for finite mixture modelling. The results confirm that our proposal improves both estimation accuracy and computational efficiency, consistently outperforming existing packages. We also demonstrate that finite mixture models can approximate analytical reliability solutions with fewer components than the actual number of system components. Our proposals are also validated using a practical example from Backblaze hard drive data. All improvements are included in the open-source rebmix R package, with complete source code provided to support the broader adoption of the R programming language in reliability analysis.

Keywords: system reliability; mixture model; numerical modelling; density estimation; parameter estimation; EM algorithm; REBMIX algorithm (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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