Probability & Statistics Day, 2013

University of California, San Diego, Friday, April 5th, 2013

Peter Hall (University of Melbourne and UC Davis)

Distribution approximation, Roth's Theorem, and looking for insects in shipping containers. [slides]

Methods for distribution approximation, including the bootstrap, do not perform well when applied to lattice-valued data. For example, the inherent discreteness of lattice distributions confounds both the conventional normal approximation and the standard bootstrap when used to construct confidence intervals. However, in certain problems involving lattice-valued random variables, where more than one sample is involved, this difficulty can be overcome by ensuring that the ratios of sample sizes are quite irregular. For example, at least one of the ratios of sample sizes could be a reasonably good rational approximation to an irrational number. Results from number theory, in particular Roth's theorem (which applies to irrational numbers that are the roots of polynomials with rational coefficients), can be used to demonstrate theoretically the advantages of this approach. This project was motivated by a problem in risk analysis involving quarantine searches of shipping containers for insects and other environmental hazards, where confidence intervals for the sum of two binomial proportions are required.


Peter Hall is an Australian Research Council Laureate Fellow at the Department of Mathematics and Statistics, University of Melbourne, and also holds a fractional appointment as Professor of Statistics at the University of California, Davis. Originally from Australia, Peter Hall earned his D.Phil. degree from the University of Oxford in 1976. He then returned to Australia, first to the University of Melbourne before taking, in 1978, a position at the Australian National University. In November 2006, he moved back to the University of Melbourne. Peter Hall is among the world's most prolific and highly-cited authors in both probability and statistics. His research in statistics has included contributions to nonparametric statistics, in particular curve estimation and resampling: the bootstrap method, smoothing, density estimation, and bandwidth selection. He has worked on numerous applications across fields of economics, engineering, physical sciences, and biological sciences. In probability theory, he has made contributions to limit theory, spatial processes, and stochastic geometry. His awards include the Lyle and Hannan Medals, the George Szekeres Medal, the Guy Medal in Silver and most recently, the Samuel Wilks award. Peter Hall is a Fellow and former President of the Institute of Mathematical Statistics, a former President of the Bernoulli Society, a Fellow of the Australian Academy of Science, and the Royal Society.