Some contributions to the analysis of skew data on the line and circle

Pewsey, Arthur Richard (2002). Some contributions to the analysis of skew data on the line and circle. PhD thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.0000fbf5

Abstract

In the first part of this thesis we consider the skew-normal class of distributions on the line and its limiting general half-normal distribution. Inferential procedures based on the methods of moments and maximum likelihood are developed and their performance assessed using simulation. Data on the strength of glass fibre and the body fat of elite athletes are used to illustrate some of the inferential issues raised.
The second part of the thesis is devoted to a consideration of the analysis of skew circular data. First, we derive the large-sample distribution of certain key circular statistics and show how this result provides a basis for inference for the corresponding population measures.
Next, tests for circular reflective symmetry about an unknown central direction are investigated. A large-sample test and computer intensive variants of it are developed, and their operating characteristics explored both theoretically and empirically. Subsequently, we consider tests for circular reflective symmetry about a known or specified median axis. Two new procedures are developed for testing for symmetry about a known median axis against skew alternatives, and their operating characteristics compared in a simulation experiment with those of the circular analogues of three linear tests. On the basis of the results obtained from the latter, a simple testing strategy is identified. The performance of the tests against rotation alternatives is also investigated. Throughout, the use of the various tests of symmetry is illustrated using a wide range of circular data sets.
Finally, we propose the wrapped skew-normal distribution on the circle as a potential model for circular data. The distribution’s fundamental properties are presented and inference based on the methods of moments and maximum likelihood is explored. Tests for limiting cases of the class are proposed, and a potential use of the distribution is illustrated in the mixture based modelling of data on bird migration.