# Monitoring coherent systems: exact and computational statistical inference

- Überwachung kohärenter Systeme: Exakte und computerbasierte statistische Inferenzverfahren

Hermanns, Marius Lukas; Cramer, Erhard (Thesis advisor); Kamps, Udo (Thesis advisor)

*Aachen (2019)*

Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2019

Abstract

In this thesis, several models of component and system lifetimes are discussed. We differentiate between continuous monitoring of a system, where the lifetimes of all relevant components are observed, and system lifetime data. Since continuous monitored coherent systems are connected to progressive Type-II censoring, we present known preliminary results about coherent systems, progressive censoring and continuous monitoring in Part I (Chapter 1-4). In Part II, we introduce new continuous monitoring models and derive distribution results, parametric estimators and computation methods. First, we consider a new model called stopped monitoring by stopping number, where a maximum number of observed component failures is prefixed. Similarly, we introduce stopped monitoring by stopping time, where the continuous monitoring process stops at the latest at a previously chosen threshold time. Such an intervention can be due to restrictions like costs and capacity. To distinguish, continuous monitoring until the system failure is called perfect monitoring since we have a "perfect" sample with all lifetimes of relevant components. Further, we discuss various distribution structures of components like independent identically distributed (IID), independent non-identically distributed (INID), dependent non-identically distributed (DNID), and failure-dependent. Continuous monitoring was introduced with two different settings: an observation with complete information (CI) about which failure time belongs to which component; and a setup without this information called incomplete information (ICI). The continuous monitoring models, distribution structures of components and information models are combined to triple models. In Chapter 5, we show that stopped monitoring by stopping number with the IID model is connected to right censoring of a progressive Type-II censoring sample. For the CI model, we apply known results about likelihood inference and confidence intervals for exponential and Weibull distributions. For the ICI model and exponential distributions, we add a fixed-point iteration procedure to compute the maximum likelihood estimator (MLE) to a known EM algorithm approach. In Chapter 6, we present a link of stopped monitoring by stopping time with the IID model and Type-I progressive hybrid censoring. For the CI model, we discuss likelihood inference and illustrate confidence intervals for exponentially and Weibull distributed component lifetimes. Again, we derive an EM algorithm approach and a fixed-point iteration method to compute the MLE of an exponential distribution for the ICI model. Chapter 7 is about a combination of perfect monitoring and the INID model. We establish distribution results for the observed data. Moreover, we discuss likelihood inference and present a method of moments approach to estimate the scale parameter of exponential distributions. MLEs only exist under the condition that at least one component lifetime is observed for every unknown parameter. In Chapter 8, we present two models for dependence of components. First, distribution results are established for perfect monitoring with failure-dependent components. The model is connected to sequential order statistics and several results about likelihood inference are adaptable. As a second model, perfect monitoring with the DNID component lifetimes is considered. The dependence of components is modelled by a copula. We analyzed parametric inference for Archimedean copulas (Clayton family) and applied several estimation methods like maximum likelihood estimation, Kendall’s tau procedure, estimation method of inference functions for margins and canonical maximum likelihood estimation. For Farlie-Gumbel-Morgenstern copulas, maximum-likelihood-based approaches are not suitable for parametric inference. In this case, Kendall’s tau method is a reasonable choice to estimate an unknown dependence parameter as it is not based on any likelihood equation. In Part III, we consider different setups with system lifetimes. Type-I progressive hybrid censoring of system lifetimes is analyzed in Chapter 9. We illustrate fixed-point procedures for exponential distributions based on non-censored system lifetimes, and based on progressively Type-II censored k-out-of-n system lifetimes. For system lifetimes, further EM algorithm approaches are given in Chapter 10. Thereby, we interpret observed system lifetimes as an incomplete sample from a continuous monitoring process. Finally, we discover a link between continuous monitoring, which is discussed in Part II, and system lifetime data, which is the topic of Part III. Throughout the thesis, simulations underline the advantage of the fixed-point iterations and EM algorithm approaches w.r.t. convergence and computation time in comparison to the Newton-Raphson method. Moreover, the convergence of the fixed-point iterations is proved under certain conditions.

### Identifier

- DOI: 10.18154/RWTH-2019-09683
- RWTH PUBLICATIONS: RWTH-2019-09683