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Thesis (M.S.)--Brigham Young University. Dept. of Civil and Environmental Engineering, 2006. / Includes bibliographical references (p. 75-77).

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published_or_final_version / Civil Engineering / Doctoral / Doctor of Philosophy

The finite mixed Weibull distribution is an appropriate distribution in modeling the lifetime of the units having more than one possible failure cause. Due to the lack of a systematic statistical procedure of fitting the distribution to a data set, it has not been widely used in lifetime data analyses. Many areas on this subject have been studied in this research. The following are the findings and contributions. Through a change of variable, 5 parameters in a two Weibull mixture can be reduced to 3. A parameter'vector (p₁, η, β) defines a family of two-Weibull mixtures which have common characteristics. Numerous probability plots are investigated on Weibull probability paper (WPP). For a given p₁ the η-β plane is partitioned into seven regions which are labeled by A through F and S. The Region S represents the two Weibull mixtures whose cdf curves are very close to a straight line. The Regions A through F represent six typical shapes of the cdf curves on WPP, respectively. The two-Weibull mixtures in one region have similar characteristics. Three important features of the two-Weibull mixture with well separated subpopulations are proved. Two existing methods for the graphical estimation of the parameters are discussed, and one is recommended over the other. The EM algorithm is successfully applied to solve the MLE for mixed Weibull distributions when m, the number of subpopulations in a mixture is known. The algorithms for complete, censored, grouped and suspended samples with non-postmortem and postmortem failures are developed accordingly. The developed algorithms are powerful, efficient and they are insensitive to the initial guesses. Extensive Monte Carlo simulations are performed. The distributions of the MLE of the parameters and of the reliability of a two Weibull mixture are studied. The MLEs of the parameters are sensitive to the degree of separation of the two subpopulation pdfs, but the MLE of the reliability is not. The generalized likelihood ratio (GLR) test is used to determine m. Under H₀: m=1 and H₁: m=m₁>1, ζ, the GLR is independent of the parameters in the distribution of H₀. The distributions of ζ or -21n(ζ) with n=50, 100 and 150 are obtained through Monte Carlo simulations. Compared with the chi-square distribution, they fall in between x²(4) and x²(6), and they are very close to x²(5). A FORTRAN computer program is developed to conduct simulation of the GLR test for 1 ≤ m₀ < m₁ ≤ 5.

by Siu-Keung Wan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 69-70). / Chapter Chapter 1 --- Introduction and Review / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- Review --- p.3 / Chapter Chapter 2 --- A general IRR model for a system with unobservable states / Chapter 2.1 --- Model and assumptions --- p.8 / Chapter 2.2 --- Long-run average cost per unit time incurred in a replacement cycle --- p.14 / Chapter Chapter 3 --- The Algorithm / Chapter 3.1 --- The key point and the local turning point of the average cost --- p.16 / Chapter 3.2 --- A finite algorithm --- p.23 / Chapter Chapter 4 --- Numerical examples and sensitivity analysis with discussion / Chapter 4.1 --- Weibull Distribution Case --- p.25 / Chapter 4.2 --- Gamma Distribution Case --- p.28 / Chapter 4.3 --- Exponential Distribution Case --- p.30 / Chapter 4.4 --- Sensitivity analysis --- p.32 / Chapter 4.5 --- Conclusion and further development --- p.38 / Figures --- p.40 / Tables --- p.46 / References --- p.69

Ng Hon Keung Tony. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 85-87). / Abstracts in English and Chinese. / Chapter Chapter 1 --- Introduction --- p.1 / Chapter Chapter 2 --- Probability Analysis of Consecutive-k-out-of-n: F system with (k-l)-step Markov Dependence --- p.11 / Chapter 2.1 --- Model and Assumptions --- p.11 / Chapter 2.2 --- Find Out the Failure Risk of the System --- p.17 / Chapter 2.3 --- The Linear Consecutive-3-out-of-4: F Repairable system with 2-step Markov Dependence --- p.27 / Chapter 2.3.1 --- Failure risk of the system --- p.27 / Chapter 2.3.2 --- Priority repair rule --- p.32 / Chapter 2.4 --- The Circular Consecutive-3-out-of-4: F Repairable system with 2-step Markov Dependence --- p.41 / Chapter 2.4.1 --- Failure risk of the system --- p.41 / Chapter 2.4.2 --- Priority repair rule --- p.47 / Chapter Chapter 3 --- System Analysis for Linear Consecutive-3-out-of-4: F system with 2-step Markov Dependence by Laplace Transform Method --- p.57 / Chapter Chapter 4 --- Reliability Indices for Consecutive-3-out-of-4: F system with 2-step Markov Dependence by Numerical Method and Simulation Study --- p.69 / Chapter 4.1 --- Numerical Method --- p.69 / Chapter 4.1.1 --- Linear consecutive-3-out-of-4: F repairable system with 2-step Markov dependence --- p.70 / Chapter 4.1.2 --- Comparison between Laplace transform method and numerical method --- p.72 / Chapter 4.1.3 --- Circular consecutive-3-out-of-4: F repairable system with 2-step Markov dependence --- p.75 / Chapter 4.2 --- Numerical Results --- p.75 / Chapter 4.3 --- Simulation Study --- p.78 / Chapter 4.4 --- Comparison between Simulation Method and Numerical Method --- p.79 / FIGURES 1 to 7 --- p.82-84 / REFERENCES --- p.85-87

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