Robustness Theory and Application
Robustness Theory and Application is an important resource for all statisticians interested in the topic of robust statistics. This book encompasses both past and present research, making it a valuable supplemental text for graduate-level courses in robustness. Brenton R. Clarke, PhD is an experienced academic in Mathematics and Statistics at Murdoch University, Perth, WA, Australia. A former president of the Western Australian Branch of the Statistical Society of Australia, Dr. Clarke has published numerous journal articles in his areas of research interest, which include linear models, robust statistics, and time series analysis.
Robustness Theory and Application
It could be said that the genesis of this book came out of a unit which was on robust statistics and taught by Noel Cressie in 1976 at Flinders University of South Australia. Noel's materials for the lectures were gathered from Princeton University where he had just completed his PhD. Having been introduced to M-, L-, and R-estimators I shifted to the Australian National University in 1977 to work on the staff at the Statistics Department in the Faculties. There I enrolled part time in a PhD with Professor C. R. Heathcote (affectionately known as Chip by his colleagues and family) who happened to be researching the integrated squared error method of estimation and more generally the method of minimum distance. The common link between the areas of study, robust statistics and minimum distance estimation, was that of M- estimation. Some minimum distance estimation methods can be represented by M-estimators. A typical model used in the formulation of robustness studies was the "epsilon-contaminated normal distribution." In the spirit of John W. Tukey from Princeton University the relative performance of the estimator, usually of location, was to consider it in such contaminated models. It occurred to me that one could also estimate the proportion of contamination, epsilon, in such models and when I proposed this to Chip he became enthusiastic that I should work on these mixture models for estimation in my PhD. Chip was aware that the trend for PhDs was to have a motivating set of data and to this end he introduced me to recently acquired earthquake data recordings which could be modeled with mixture modeling. A portion of a large data set was passed on to me by Professor R.S. Anderssen (known as Bob), also at the Australian National University. Bob also introduced me to the Fortran Computing Language. My brief was to compute minimum distance estimators on the earthquake data. In the mean time, Chip introduced me to Professor Frank Hampel's PhD thesis and several references on mixture modeling. After 1 year of trying to compute variance covariance matrices for the minimum distance estimation methods and for some reason failing to get positive definite matrices as was expected, I decided to come back to M-estimation and study the theory more closely. An idea germinated that I could study the M-estimator at a distribution other than at the model parametric family and other than at a symmetric contaminating distributions. This became the inspiration for my own PhD work.
I had the good fortune to then cross paths with Peter Hall. Chip who had been burdened with the duties as Dean of the Faculty of Economics at ANU took sabbatical at Berkeley for a year and Peter became my supervisor. Peter was always so cheerful and encouraging when it came to research. He was publishing a book on the "Martingale limit theory and its application" with Chris Heyde, and he encouraged me to read books and papers on limit theorems. I thus became interested in the calculus associated with limit theorems, and asymptotic theory of M-estimators. Chip returned to ANU in 1980 and kindly advised me on the presentation of my thesis and arranged for three quality referees, one of whom was Noel Cressie!
For some reason I wanted to go overseas and see the world. This was made possible with a postdoctoral research assistant position to study time series analysis at Royal Holloway College, University of London, in the period 1980-1982. While I worked on time series, I took my free time to put together my first major publication. Huber's (1981) monograph had come out. My paper was to illustrate that for a large class of statistics that could be represented by statistical functionals, which were in fact M-estimators, it was possible to inherit both weak continuity and Fréchet differentiability. These qualities in turn provide inherent robustness of the statistics. From the time of first submission to actual publication in