MSI Weekly Bulletin - Week starting Monday 27 February, 2006

Both parametric distribution functions appearing in extreme value theory - the generalized extreme value distribution and the generalized Pareto distribution - have log - concave densities if the extreme value index $\gamma\in[-1,0]$. It is proved that all distribution functions $F$ having a log - concave density function belong to the max - domain of attraction of the generalized extreme value distribution. Replacing the order statistics in tail index estimators by their corresponding quantiles from the smoothed estimated distribution function that is based on the estimated log - concave density $\hat f_n$ leads to novel smoothed semi-parametic tail index estimators. Monte Carlo simulations suggest that these estimators lead to highly accurate estimators for $/gamma$ and are well superior to their non-smoothed counterparts.

Modelling time-dependent materials, both solids and fluids, is of growing importance for today\'s and tomorrow\'s consumer society. Polymer products are everywhere: in the home, in the office, on our roads and in the skies. Yet different polymers degrade or deform with time in different ways. Consequently domestic appliances break down, laptop lids crack, plastic coatings split, and airtight seals leak. If industrial manufactureres could predict accurately the natural lifetime of a product, or better still, predict its behaviour over long time periods, they could save billions of dollars worldwide in servicing and maintenance costs. The mathematical modelling of materials with memory draws upon experimental observations made on simple flows or deformations of the material. The way in which materials wobble, creep and relax gives important information on the molecular microstructure. Parameter estimation is of paramount importance, and there is a rich abundance of difficult inverse problems which are severely ill-posed. In this talk, I will describe some of the mathematical beauty and challenges of the subject.