Statistical Problems Relevant to Risk Assessment

Abstract

This thesis discusses two theoretical statistical problems which have potential uses in risk assessment. The likely application of the first problem is to ecotoxicological
risk assessment, while the second problem has a wide range of potential applications in risk assessment. The two problems have important mathematical features in common.
The first problem is concerned with key dominance properties for the arithmetic mean as the sample size increases. We show mathematically that the dominance properties hold for all distributions with symmetric log-concave densities. A detailed and comprehensive analysis of what happens when the sample size increases from one to two for two-component scale and location mixtures of normal distributions is introduced.
The second problem relates to combining limited probabilistic expert judgements on multiple quantities in order to provide limited probabilistic information about
a derived quantity. First, a working hypothesis that simplifies calculations for the derived quantity is developed. Second, we mathematically show that the working
hypothesis holds for all distributions with symmetric log- concave densities. In addition, it holds for negatively-skewed Azzalini-style skew-symmetric distributions with log-concave kernels when two quantities are involved. Moreover, under a specific condition, the working hypothesis is valid whenever the underlying distributions have log-concave right tail probability functions or partial log-concave right tail probability functions when two quantities are involved.