Abstracts for 2015 Challis Lectures
by Jon Wellner
General Lecture (Dec 10, 2015)
Empirical Distributions and Empirical Processes: a Short History (video)Empirical distributions have a long history in probability and statistics, starting possibly with Galton and proceeding through the work of Kolmogorov, Glivenko, and Cantelli in the 1930's. The contributions of J. L. Doob and Monroe Donsker in the late 1940's and early 1950's added statistical content and depth to the early studies. Nevertheless, most of the early history and development concerned empirical distributions of real-valued random variables. This began to change in the 1960's and 1970's, but progress was slow until the ground breaking work of Vapnik and Chervonenkis in the late 1960's and early 1970's, and R. M. Dudley in the mid 1970's. Their contributions enabled a formulation which allows for arbitrary sample spaces. The theoretical and practical developments accelerated in the 1980's resulting in some very beautiful and statistically important developments concerning validity of the bootstrap. In this lecture I will review the history of empirical distributions and processes with a view toward the interactions with statistical questions and issues. I will also formulate some problems and open questions which I believe will be important for the future.
Technical Lecture (Dec 11, 2015)
Nonparametric Estimation of Log-Concave and s-Concave Densities: an Alternative to Maximum Likelihood (video)The class of log-concave densities on R^d provides a natural nonparametric generalization of the class of multivariate normal distributions. The log concave class is preserved by many familiar operations, including affine transformations, convolution, marginalization, and weak limits. On the other hand, the log-concave class only include densities with sub-exponential tails. The s-concave classes (for s<0) give classes which are progressively larger as s decreases to negative infinity and which include the (multivariate) t-densities. In this talk I will discuss maximum likelihood estimation (MLE) of log-concave and s-concave densities on R and R^d. I will review some of the properties of these estimators, both good and bad. I will also discuss recent results for the Rényi divergence estimators proposed by Koenker and Mizera (2010). (This talk is based on joint work with Charles Doss, Qiyang (Roy) Han, Fadoua Balabdaoui, and Arseni Seregin as well as that of others working in this area, including Richard Samworth and Lutz Dümbgen.)