Statistical inference for the optimal approximating model

In the setting of high-dimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive (point) estimation, the construction of adaptive confidence regions is severely limited (cf. Li in Ann Stat 17:1001–1008, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence regions for the best approximating model in terms of risk. One of our constructions is based on a multiscale procedure and a particular coupling argument. Utilizing exponential inequalities for noncentral χ2-distributions, we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one.

Stereological Modelling of Random Particles

Stochastic models for three-dimensional particles have many applications in applied sciences. Lévy–based particle models are a flexible approach to particle modelling. The structure of the random particles is given by a kernel smoothing of a Lévy basis. The models are easy to simulate but statistical inference procedures have not yet received much attention in the literature. The kernel is not always identifiable and we suggest one approach to remedy this problem. We propose a method to draw inference about the kernel from data often used in local stereology and study the performance of our approach in a simulation study.

Climate refugia: joint inference from fossil records, species distribution models and phylogeography

Climate refugia, locations where taxa survive periods of regionally adverse climate, are thought to be critical for maintaining biodiversity through the glacial–interglacial climate changes of the Quaternary. A critical research need is to better integrate and reconcile the three major lines of evidence used to infer the existence of past refugia – fossil records, species distribution models and phylogeographic surveys – in order to characterize the complex spatiotemporal trajectories of species and populations in and out of refugia. Here we review the complementary strengths, limitations and new advances for these three approaches. We provide case studies to illustrate their combined application, and point the way towards new opportunities for synthesizing these disparate lines of evidence. Case studies with European beech, Qinghai spruce and Douglas-fir illustrate how the combination of these three approaches successfully resolves complex species histories not attainable from any one approach. Promising new statistical techniques can capitalize on the strengths of each method and provide a robust quantitative reconstruction of species history. Studying past refugia can help identify contemporary refugia and clarify their conservation significance, in particular by elucidating the fine-scale processes and the particular geographic locations that buffer species against rapidly changing climate.