| Robust techniques are becoming a core component of statistical practice and they have considerable support in statistical research literature, both at a highly abstract mathematical level and in extensive Monte Carlo studies.;In this dissertation we propose a robust analysis for the Delta distribution which is widely used in many practical fields. It is a mixture of a degenerate distribution at zero and a highly skewed distribution, namely, the lognormal distribution. General methods are presented with focus on applications in marine sciences.;Through the relationship between the normal and the lognormal distributions, we start with a robust estimation for the parameters of the Delta distribution. Based on the suggested robust parameter estimators and in an attempt to find robust estimates of the mean and variance of the Delta distribution based on existing optimal estimators, we investigate a notion we call "the invariance property of robustness". That is, are functions of robust estimators robust?;The second part of the dissertation deals with robust analysis of variance for the Delta distribution. The classical testing methods for comparing independent groups are considered optimal when the associated assumptions are adhered to. When assumptions are violated, however, they give good results up to a point, but eventually the performance of the tests becomes unsatisfactory. The discontinuity and extreme skewness of the Delta model are added complexities that cause conclusions based on standard procedures to be misleading. The discontinuity can be handled by considering separate analysis for the proportion of zeros and the extreme non-normality can be handled using bootstrap testing methods based on a robust version of the classical test statistic by comparing robust measures of location. The results from the two tests can be combined adjusting the significance probability using a bootstrap-based multiple testing procedure.;Finally, we present an example from marine sciences to demonstrate some aspects of the proposed testing procedures. |
