A simulation-based reliability analysis method is presented and evaluated. This method is intended for problems for which most probable point of failure (MPP) search-based methods fail or provide inaccurate results, and for which Monte Carlo simulation and its variants are too costly to apply. This may occur in the evaluation of complex engineering problems of low failure probability. The method used to address this problem is a variant of conditional expectation and works by sampling on the failure boundary without relying on the MPP. The effectiveness of the method is compared to a selection of other commonly available reliability methods considering a variety of analytical as well as more complex engineering problems. The results indicate that the method has the potential to deliver solutions of high efficiency and accuracy for a wide range of difficult reliability problems.

This thesis develops a methodology for the construction of generalized families of probability distributions in the quantile statistical universe, that is, distributions specified in terms of their quantile functions. The main benefit of the proposed methodology is that it generates quantile-based distributions with skewness-invariant measures of kurtosis. The skewness and kurtosis can therefore be identified and analyzed separately. The key contribution of this thesis is the development of a new type of the generalized lambda distribution (GLD), using the quantile function of the generalized Pareto distribution as the basic building block (in the literature each different type of the GLD is incorrectly referred to as a parameterization of the GLD – in this thesis the term type is used). The parameters of this new type can, contrary to existing types, easily be estimated with method of L-moments estimation, since closed-form expressions are available for the estimators as well as for their asymptotic standard errors. The parameter space and the shape properties of the new type are discussed in detail, including its characterization through L-moments. A simple estimation algorithm is presented and utilization of the new type in terms of data fitting and approximation of probability distributions is illustrated. / Thesis (PhD)--University of Pretoria, 2013. / gm2014 / Statistics / unrestricted