As a mathematician I worked on the construction of prime representing polynomials with Jim Jones et al.; this was an area opened up by the (negative) solution to Hilbert's Tenth Problem given by Yuri Matiyasevich. (This earned me an Erdös number of 3.) I then studied for and completed some entry level examinations of the Society of Actuaries, thus becoming interested in Statistics.
Applied Probability; Distribution
Theory; Engineering
Applications
Some of my first work in Statistics concerned applications of probability to such fields as reliability theory (1, 2, 3) and queueing theory, and orderings of lifetime distribution functions (with Subhash Kochar). Random Fibonacci sequences were studied with Dalhousie University colleagues George Gabor, Robert Dawson and Richard Nowakowski. A paper on pattern reduction matrices in statistics led to a number of applications in multivariate analysis (1, 2), in distribution theory and in nonlinear regression (with David Hamilton). Improved method of moments estimators for the gamma distribution have been obtained (with Julian Cheng and Norman Beaulieu), and the distribution of the sum of all but the smallest few order statistics has been studied, with Pavel Loskot and Norman Beaulieu. Optimal (but not necessarily robust) designs in various situations have been obtained (1, 2) with Zhide Fang and his colleagues. With Oleg Michailovich I have looked at some problems involving the separation of signal sources. Jesús López-Fidalgo, Raul Martín-Martín and I studied marginally restricted D-optimal designs.
Much of my work has been devoted to the derivation of statistical methods which are robust, i.e. not highly sensitive to departures from the assumptions under which they are derived. Early work in this area concentrated on estimation methods for multivariate location and scatter (with Z. Zheng of Peking University). This developed into a systematic search, with John Collins, for minimax robust estimators (1, 2, 3) , i.e. estimators whose maximum asymptotic variance, as the distribution of the data ranges over a particular class, is a minimum. Related work, with Myron Hlynka and Jerome Sheahan, concerned adaptively optimal robust estimation. To this point I had largely concentrated on M-estimation (a robustification of maximum likelihood methods) of location or regression. I then began to consider L- and R-estimation (1, 2, 3, 4) and scale estimation (1, 2) (together with graduate students Eden Wu, Yu Wang and Julie Zhou) as well as minimum distance estimation. Optimal procedures were developed in each case. As well, computational techniques in robust estimation, asymptotics of GM-estimation and (with my postdoctoral fellow Sanjoy Sinha) minimax weights for GM-estimation were studied. Robust regression methods for dependent data (with Chris Field) have been proposed and implemented, as have jackknife-based diagnostics for robust regression (with my graduate student Zhiyi Du) and robust methods of prediction in spatial studies.
For some years now I have been concerned with the derivation of
regression designs which are robust against violations of various model
assumptions - linearity of the response (1, 2, 3, 4,
5),
independence
(1, 2,
both with Julie
Zhou) and homoscedasticity
of the errors, etc. These are again minimax procedures,
with respect to a (loss) function of the mean squared error of the
predicted
values. Three new approaches to design theory - the infinitesimal
approach for robustness against dependent errors (with Julie
Zhou), the minimum variance unbiased (1,
2, 3)
approach for weighted regression, and the minimum
average loss approach (with Adeniyi
Adewale) - have been developed. With graduate students and other
collaborators I have investigated extrapolation designs
(1,
with Zhide
Fang, 2, 3,
with Xiaojian Xu),
wavelet designs
(1, 2,
both with Alwell
Oyet), restricted
minimax designs (Giseon
Heo, Byron
Schmuland) and designs
for approximate logistic models (with Adeniyi
Adewale). Polynomial
designs have been studied with Shawn
Liu; this work was extended,
by applying the theory of canonical
moments, with Zhide
Fang.
Methods of simulated
annealing have been introduced to robust design theory (with Zhide
Fang). Robust
designs for clinical
trials and for nonlinear
regression (with Sanjoy
Sinha, with whom the asymptotic
aspects were also investigated) have been studied, as have robust
methods of design
in spatial studies and, with Julie
Zhou, field
experiments. Holger Dette
and I have studied designs for 3D
shape analysis and for series
estimation. This collaboration motivated a study of the
asymptotic properties of a Neyman-Pearson
test for model discrimination; these results have been applied to the
construction of robust
discrimination designs, and designs for the testing of lack of fit in binary response
models. For designs appropriate for M-estimation see 1,
and, with Eden Wu, 2. Designs for dose-response studies have been
investigated, with Pengfei
Li.
Applied Statistics; Data Analysis
These and other methods have been applied in Forestry (1, 2, 3, with Stephen Titus, Shongming Huang and Mike Bokalo) and in Oceanography (with John Haines and Keith Thompson). Work with Zack Florence and Michelle Hiltz from the Alberta Research Council has led to the development of models, with associated software and manual for the estimation of source contributions to airborne particulate matter.