Upper bounds for spatial point process approximations

We consider the behavior of spatial point processes when subjected to a class of linear transformations indexed by a variable T. It was shown in Ellis [Adv. in Appl. Probab. 18 (1986) 646-659] that, under mild assumptions, the transformed processes behave approximately like Poisson processes for large T. In this article, under very similar assumptions, explicit upper bounds are given for the d_2-distance between the corresponding point process distributions. A number of related results, and applications to kernel density estimation and long range dependence testing are also presented. The main results are proved by applying a generalized Stein-Chen method to discretized versions of the point processes.Comment: Published at http://dx.doi.org/10.1214/105051604000000684 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Demand for Enhanced Annuities

In enhanced annuities, the annuity payment depends on one's state of health at some contracted date while in "standard annuities", it does not. The focus of this paper is on an annuity market where "standard" and enhanced annuities areoffered simultaneously. When all insured know equally well on their future health status either enhanced annuities drive standard annuities out of the market or vice versa. Both annuity types can exist simultaneously when the insured know varying exactly on their risk type. In the case of the existence of such an "interior" solution, its is derived that this solution must be unique in the case of risk averse insured and that it Pareto-dominates the corner solution. Finally, it is shown that in all cases where at least part of the insured buy enhanced annuities social welfare is reduced

Stein's method and Poisson process approximation for a class of Wasserstein metrics

Based on Stein's method, we derive upper bounds for Poisson process approximation in the $L_1$-Wasserstein metric $d_2^{(p)}$, which is based on a slightly adapted $L_p$-Wasserstein metric between point measures. For the case $p=1$, this construction yields the metric $d_2$ introduced in [Barbour and Brown Stochastic Process. Appl. 43 (1992) 9--31], for which Poisson process approximation is well studied in the literature. We demonstrate the usefulness of the extension to general $p$ by showing that $d_2^{(p)}$-bounds control differences between expectations of certain $p$th order average statistics of point processes. To illustrate the bounds obtained for Poisson process approximation, we consider the structure of 2-runs and the hard core model as concrete examples.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ161 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

The History of Bees by Maja Lunde

Book review of Maja Lunde\u27s The History of Bees