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Open Research Online Oro.Open.Ac.Uk Open Research Online The Open University’s repository of research publications and other research outputs Some contributions to the analysis of skew data on the line and circle Thesis How to cite: Pewsey, Arthur Richard (2002). Some contributions to the analysis of skew data on the line and circle. PhD thesis The Open University. For guidance on citations see FAQs. c 2002 Arthur Richard Pewsey Version: Version of Record Link(s) to article on publisher’s website: http://dx.doi.org/doi:10.21954/ou.ro.0000fbf5 Copyright and Moral Rights for the articles on this site are retained by the individual authors and/or other copyright owners. For more information on Open Research Online’s data policy on reuse of materials please consult the policies page. oro.open.ac.uk S o m e C ontributions t o t h e A n a l y s is o f S k e w D a t a o n t h e L in e a n d Cir c l e by Arthur Richard Pewsey b s c , m sc Thesis submitted for the degree of Doctor of Philosophy D epartm ent of S tatistics F aculty of M athematics and S tatistics T he O p e n U n iversity O c t o b e r 2oo2 of" SOBA^SSX^A/'. 9 2,OOZ, b a re oF ^ ProQuest Number: C811633 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely ev e n t that the author did not send a co m p le te manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest C811633 Published by ProQuest LLO (2019). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States C o d e Microform Edition © ProQuest LLO. ProQuest LLO. 789 East Eisenhower Parkway P.Q. Box 1346 Ann Arbor, Ml 4 8 1 0 6 - 1346 A bstract 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. This thesis is dedicated with great respect, gratitude and affection to P rofessor Toby L ew is Acknowledgements As I have grown older, I have become increasingly more reflective upon my past. The physical distance presently separating me from my roots only heightens the competing senses of melancholy, for what has been, and deep gratitude to those from whom I have learnt. Sadly, it is too late to express in person my gratitude to some who are most deserving of it. Nevertheless, I would like to take this opportunity to express the sincere gratitude I feel to those who have directly contributed towards my knowledge and understanding of Mathematics and Statistics, of which this thesis is an expression. Going back in time at least 25 years, two people who influenced my mathematical development greatly were my father and my ‘A’ level mathematics teacher Mr Derek Friggins. My father skillfully used evening washing up sessions to instill in me basic concepts of algebra. On deriving the theoretical results of Chapter 3, I was overcome by sensations similar to those I had felt on obtaining beautifully simple solutions to some initially horrendous problems in Applied Mathematics involving interminable trigonometric expressions. The late Mr Friggins was the person who set those problems and taught me the mathematics required to solve them. Moving on to my years as a university student in Hull, I am infinitely grateful to those who extended my mathematical knowledge and introduced me to the world of Statistics. I received my first course in Statistics from Edward Evans, and he and Michael Bingham kindled my initial interest in things statistical. In my second and third years, the stimulating lectures and tutorials given by Michael Bingham, Carolyn Craggs, Fergus Daly, Michael Goldstein, Wilfred Kendall and Toby Lewis heightened that interest still further. For the influence they had on my wider understanding of Statistics whilst a research student in Newcastle, I would like to express my gratitude to the late Jon Anderson, Robin Plackett and, especially, Peter Diggle. I would also like to thank my former colleagues Michael Bingham, Michael Goldstein and Jim Thompson, in Hull, and Ray Harris, Martin Roberts, Tony Scallan and Chris Sutton, in Preston, for their multifarious contributions to my greater knowledge and understanding of Statistics. However, this thesis is also a testament to the help and support of many other people. I am greatly indebted to Kevin McConway who, whilst Head of the Department of Statistics, saw through my application to study as a research ill student. I thank both him and his successor, Paul Garthwaite, for co-ordinating the administrative and managerial requirements of my studies. Regarding the other forms of support I have received during the research which led to the production of this thesis, I would like to express my sincere gratitude to: i) The Open University for its excellent distance learning support and on­ line library facilities. ii) Anunciacion Gutierrez, Encarna Pegado, Chris Sutton and my supervisors Chris Jones and Toby Lewis for tracking down sources in the literature. iii) Tony Scallan and David Wooff for pointers to software archives. iv) Bruno Bruderer of the Swiss Ornithological Station in Sempach, Switzerland, for providing me with the data analyzed in Chapter 6. v) Adelchi Azzalini for letting me have access to a pre-publication version of Azzalini & Capitanio (1999) and for stimulating communication by e- mail. vi) Leon Harter for his generous guidance to published work related to the general half-normal distribution. I have all sorts of reasons to be grateful to my wife, Lucia Aguilar Zuil. It was she who enticed me to Spain, a move which acted as a catalyst to renew my interest in research. She has been my emotional pillar of strength throughout. There are three people whose influence on the production of this thesis has been absolutely fundamental: Nick Fisher, Chris Jones and Toby Lewis. The original idea of wrapping the skew-normal distribution on to the circle so as to produce a model for skew circular data is due to Nick Fisher who posed it as a problem to my supervisor Toby Lewis to be cogitated over accompanied by an early morning brandy. All the other ideas expressed within the six chapters making up this thesis can, in fact, be traced back to that one original idea. So, thank you, Nick! The supervision I have received from Chris Jones has been truly excellent. Mostly our communication has been via e-mail and post, and from the experience of using these forms of communication I have been convinced of the efficacies of both and the enormous potential of conducting research “at a distance”. Rarely did I have to wait more than half an hour for a response to any e-mail sent by me. Usually within a week of posting the draft of a paper or a chapter of this thesis, I would receive an e-mail with concise and insightful feedback. The contents of my published papers and this thesis have benefitted enormously from that feedback. Thank you for all your hard work, Chris! IV I find it incredibly difficult to know where to start in expressing my gratitude to Toby Lewis. I have known Toby for nearly 25 years and during that time he has been an inspirational lecturer, friend, potential supervisor, role model, confidant, supervisor and guru to me. Always positive, always supportive, always young at heart. I feel immensely privileged that in 1978 our paths in life crossed. For all these reasons, and more, Toby, this thesis is dedicated to you. Thank you so very much.
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