Useful generalized properties of L1 estimators

Recent results by G. Appa and C. Smith, as well as I. Barrodale and F. D. K. Roberts, underscore several properties exhibited for fitting a linear model to a set of observation points under the criterion of least sum of absolute deviations(commonly denoted as the L₁ criterion). This paper will generalize these properties to the non-full rank case and relax in a natural way some assumptions given by Appa and Smith.     

On the asymptotic bias of grenander’s mode estimator

Grenander introduced a direct estimator of the mode for a large class of densities. This note considers a large subclass of these densities for which Grenander’s estimator is asymptotically biased. Some of the distributions from this subclass include the F, gamma, and beta for which asymptotic expressions for the bias are given. To reduce the bias, it is recommended to choose larger values for one of the parameters of the estimator when the underlying distribution is nonsymmetric.     

On least absolute values estimation

The resistance of least absolute values (L₁) estimators to outliers and their robustness to heavy-tailed distributions make these estimators useful alternatives to the usual least squares estimators. The recent development of efficient algorithms for L₁ estimation in linear models has permitted their use in practical data analysis. Although in general the L₁ estimators are not unique, there are a number of properties they all share. The set of all L₁ estimators for a given model and data set can be characterized as the convex hull of some extreme estimators. Properties of the extreme estimators and of the L₁-estimate set are considered.     

Breeding performance of an apex predator,the peregrine falcon,across urban and rural landscapes

Urban environments present wildlife with major challenges and yet surprising numbers of species have colonised towns and cities globally. Despite the growing realisation that urban centres can be important habitats for wildlife, why some species do better than others in urban environments remains poorly understood. Here, we compare the breeding performance of an apex predator, the peregrine falcon (Falco peregrinus), in urban and rural environments, and test whether variation in reproductive success between and within environments is driven by prey. Historical breeding data were collected from raptor study groups across Great Britain between 2006 and 2016, from 22 urban and 58 rural nest sites, involving 101 and 326 nesting attempts, respectively. Prey density, biomass and diversity around the individual nests was estimated using modelled estimates from a national bird census. Urban peregrines produced more fledglings and had a higher overall nesting success (i.e. whether a nesting attempt was successful or unsuccessful) than rural peregrines. Prey density and biomass were significantly higher, and diversity significantly lower, in the urban sites, and explained the variation in reproductive success within both the urban and rural environments. Therefore, urban environments in Great Britain appear to provide peregrine falcons with superior habitats in terms of prey availability compared to rural habitats. We conclude that some apex predators can benefit from urban environments and that urban planning has the potential to benefit biodiversity across many trophic levels.