On the two-phase framework for joint model and design-based inference

We establish a mathematical framework that formally validates the two-phase ``super-population viewpoint'' proposed by Hartley and Sielken [Biometrics 31 (1975) 411--422] by defining a product probability space which includes both the design space and the model space. The methodology we develop combines finite population sampling theory and the classical theory of infinite population sampling to account for the underlying processes that produce the data under a unified approach. Our key results are the following: first, if the sample estimators converge in the design law and the model statistics converge in the model, then, under certain conditions, they are asymptotically independent, and they converge jointly in the product space; second, the sample estimating equation estimator is asymptotically normal around a super-population parameter.Comment: Published at http://dx.doi.org/10.1214/009053605000000651 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Stochastic Gross-Pitaevskii Methodology

We review the stochastic Gross-Pitaevskii approach for non-equilibrium finite temperature Bose gases, focussing on the formulation of Stoof; this method provides a unified description of condensed and thermal atoms, and can thus describe the physics of the critical fluctuation regime. We discuss simplifications of the full theory, which facilitate straightforward numerical implementation, and how the results of such stochastic simulations can be interpreted, including the procedure for extracting phase-coherent (`condensate') and density-coherent (`quasi-condensate') fractions. The power of this methodology is demonstrated by successful ab initio modelling of several recent atom chip experiments, with the important information contained in each individual realisation highlighted by analysing dark soliton decay within a phase-fluctuating condensate.Comment: Unedited version of chapter to appear in Quantum Gases: Finite Temperature and Non-Equilibrium Dynamics (Vol. 1 Cold Atoms Series). N.P. Proukakis, S.A. Gardiner, M.J. Davis and M.H. Szymanska, eds. Imperial College Press, London (in press). See http://www.icpress.co.uk/physics/p817.htm

Invariant expectation values in the sampling of discrete frequency distributions

The general relationship between an arbitrary frequency distribution and the expectation value of the frequency distributions of its samples is discussed. A wide set of measurable quantities ("invariant moments") whose expectation value does not in general depend on the size of the sample is constructed and illustrated by applying the results to Ewens sampling formula. Invariant moments are especially useful in the sampling of systems characterized by the absence of an intrinsic scale. Distribution functions that may parametrize the samples of scale-free distributions are considered and their invariant expectation values are computed. The conditions under which the scaling limit of such distributions may exist are described.Comment: arXiv admin note: substantial text overlap with arXiv:1210.141

Neutral Theory and Relative Species Abundance in Ecology

The theory of island biogeography[1] asserts that an island or a local community approaches an equilibrium species richness as a result of the interplay between the immigration of species from the much larger metacommunity source area and local extinction of species on the island (local community). Hubbell[2] generalized this neutral theory to explore the expected steady-state distribution of relative species abundance (RSA) in the local community under restricted immigration. Here we present a theoretical framework for the unified neutral theory of biodiversity[2] and an analytical solution for the distribution of the RSA both in the metacommunity (Fisher's logseries) and in the local community, where there are fewer rare species. Rare species are more extinction-prone, and once they go locally extinct, they take longer to re-immigrate than do common species. Contrary to recent assertions[3], we show that the analytical solution provides a better fit, with fewer free parameters, to the RSA distribution of tree species on Barro Colorado Island (BCI)[4] than the lognormal distribution[5,6].Comment: 19 pages, 1 figur