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Nicole Augustin
Email: N.H.Augustin@bath.ac.uk

Alex Cox
Email: A.M.G.Cox@bath.ac.uk

### Probability and Statistics Seminars: Winter 2007

This page is now historical. Details of current seminars in Probability and Statistics.

Our seminars our usually held at 2.15 p.m. on Fridays in room 3W 3.7 . If you wish to find out more, please contact one of the organisers.

** LANDSCAPES ** 9/11/07: Wilfred Kendall (Warwick)

Short-length routes in low-cost networks

Note: Seminar in Landscapes Series: 4.15 in 3WN 2.1

16/11/07: Peter Jupp (St Andrews)

Estimation of population size: conditioning has negligible effect

Many methods of estimating the size N of a homogeneous population are based on i.i.d. random variables x_1, ..., x_n (e.g. capture histories, distances from a transect), of which only a random number n are observed. Both the distribution of x_1, ..., x_n and the probability that x_i is observed depend on a (vector) parameter theta. Two appealing estimators of (N,theta) are

(a) the full m.l.e. ({\hat N},{\hat theta}),

(b) the conditional m.l.e. ({\hat N}_c,{\hat theta}_c), where {\hat theta}_c is the m.l.e. of theta obtained by conditioning on n, and {\hat N}_c is a Peterson-type estimator.

In this talk I shall describe recent work with Rachel Fewster (Auckland) which has produced

(i) a formula showing that ({\hat N},{\hat theta}) and ({\hat N}_c,{\hat theta}_c) are remarkably close,

(ii) the asymptotic distribution of ({\hat N},{\hat theta}) and ({\hat N}_c,{\hat theta}_c).

An extension to non-homogeneous populations will be indicated.

Quadrature of Lipschitz Functionals and Approximation of Distributions

We study randomized (i.e. Monte Carlo) algorithms to compute expectations of Lipschitz functionals w.r.t. measures on infinite-dimensional spaces, e.g., Gaussian measures or distribution of diffusion processes. We determine the order of minimal errors and corresponding almost optimal algorithms for three different sampling regimes: fixed-subspace-sampling, variable-subspace-sampling, and full-space sampling. It turns out that these minimal errors are closely related to quantization numbers and Kolmogorov widths for the underlying measure. For variable-subspace-sampling suitable multi-level Monte Carlo methods, which have recently been introduced by Giles, turn out to be almost optimal.
Joint work with Jakob Creutzig (Darmstadt), Steffen Dereich (Bath), Thomas Müller-Gronbach (Magdeburg).