Probability-Matching Predictors for Extreme Extremes

A location- and scale-invariant predictor is constructed which exhibits good probability matching for extreme predictions outside the span of data drawn from a variety of (stationary) general distributions. It is constructed via the three-parameter {\mu, \sigma, \xi} Generalized Pareto Distribution (GPD). The predictor is designed to provide matching probability exactly for the GPD in both the extreme heavy-tailed limit and the extreme bounded-tail limit, whilst giving a good approximation to probability matching at all intermediate values of the tail parameter \xi. The predictor is valid even for small sample sizes N, even as small as N = 3. The main purpose of this paper is to present the somewhat lengthy derivations which draw heavily on the theory of hypergeometric functions, particularly the Lauricella functions. Whilst the construction is inspired by the Bayesian approach to the prediction problem, it considers the case of vague prior information about both parameters and model, and all derivations are undertaken using sampling theory.

Elemental estimators for the Generalized Extreme Value tail

In a companion paper (McRobie(2013) arxiv:1304.3918), a simple set of `elemental' estimators was presented for the Generalized Pareto tail parameter. Each elemental estimator: involves only three log-spacings; is absolutely unbiased for all values of the tail parameter; is location- and scale-invariant; and is valid for all sample sizes $N$, even as small as $N= 3$. It was suggested that linear combinations of such elementals could then be used to construct efficient unbiased estimators. In this paper, the analogous mathematical approach is taken to the Generalised Extreme Value (GEV) distribution. The resulting elemental estimators, although not absolutely unbiased, are found to have very small bias, and may thus provide a useful basis for the construction of efficient estimators.

A dynamic view of the spread and intracellular distribution of Salmonella enterica.

The events that determine the dynamics of proliferation, spread and distribution of microbial pathogens within their hosts are surprisingly heterogeneous and poorly understood. We contend that understanding these phenomena at a sophisticated level with the help of mathematical models is a prerequisite for the development of truly novel, targeted preventative measures and drug regimes. We describe here recent studies of Salmonella enterica infections in mice which suggest that bacteria resist the antimicrobial environment inside host cells and spread to new sites, where infection foci develop, and thus avoid local escalation of the adaptive immune response. We further describe implications for our understanding of the pathogenic mechanism inside the host.