Modified weibull distributions in reliability engineering

Abstract

This thesis concerns the modeling of the Weibull family to lifetime data, studies<br>the statistical properties of the distributions, and considers the parameter<br>estimation based on a complete or censored sample. Related issues such as model<br>selection, evaluating mean residual life and burn-in time are addressed as well.<br>In our research, the modified Weibull distribution and odd Weibull distribution<br>are studied. As an important step in Weibull analysis, model characterization<br>provides insight into the properties and applicability to model data of the<br>distributions. For the distributions considered, we describe the important statistics<br>and distribution functions, both in analytical and numerical ways.<br>Parameter estimation is crucial for the model to be built and is often a difficult<br>problem, especially for distributions with more than 2 parameters. In this thesis,<br>maximum likelihood estimation is studied in detail. Several techniques regarding<br>this estimation method are proposed to simplify computation, which help look<br>into the existence and uniqueness properties of the estimators. Another estimation<br>method called Markov chain Monte Carlo is used to estimate the parameters of<br>the modified Weibull distribution and is found to outperform MLE in several<br>aspects when the prior is independent generalized uniform and the size of sample<br>data is small. A graphic parameter estimation method is proposed for the odd<br>Weibull distribution. The method is especially useful when the shape parameters<br>are negative.