In this paper a kernel estimator of the differential entropy of the mark distribution of a homogeneous Poisson marked point process is proposed. The marks have an absolutely continuous distribution on a compact Riemannian manifold without boundary. We investigate L2 and the almost surely consistency of this estimator as well as its asymptotic normality.

In this paper, we present a Markov chain Monte Carlo (MCMC) simulation algorithm for estimating parameters in the kernel density estimation of bivariate insurance claim data via transformations. Our data set consists of two types of auto insurance claim costs and exhibits a high-level of skewness in the marginal empirical distributions. Therefore, the kernel density estimator based on original data does not perform well. However, the density of the original data can be estimated through estimating the density of the transformed data using kernels. It is well known that the performance of a kernel density estimator is mainly determined by the bandwidth, and only in a minor way by the kernel. In the current literature, there have been some developments in the area of estimating densities based on transformed data, where bandwidth selection usually depends on pre-determined transformation parameters. Moreover, in the bivariate situation, the transformation parameters were estimated for each dimension individually. We use a Bayesian sampling algorithm and present a Metropolis-Hastings sampling procedure to sample the bandwidth and transformation parameters from their posterior density. Our contribution is to estimate the bandwidths and transformation parameters simultaneously within a Metropolis-Hastings sampling procedure. Moreover, we demonstrate that the correlation between the two dimensions is better captured through the bivariate density estimator based on transformed data.

Home ranges of seven Guiana dolphins (Sotalia guianensis) (Van Bénéden, 1864) were studied in the Cananéia estuary (~25°03′S 47°55′W), south-eastern Brazil. Boat-based observations were conducted from May 2000 to July 2003 in ~132 km2 of protected inner waters. The photo-identification technique was used to follow naturally marked individuals through time and space. From a total of 138 catalogued individuals, five males and two females presented 20+ sightings and were used for home range estimation. Sightings were plotted and analysed in a Geographic Information System (GIS). With the ‘Home Range Tools’ extension the fixed kernel density estimator with band width (smoothing parameter) chosen via least squares cross-validation was performed for each individual. The fixed kernel method was used to estimate the non-parametric utility distribution of each dolphin, keeping band width (h) constant for a data set. The first polygons created by these parameters had an amoeboid shape and in some cases more than one centre of activity. The 95% home range estimated outlines varied from 1.6 to 22.9 km2 (7.9 ± 8.3 km2). This large interval shows strong evidences on individual variation in S. guianensis' home ranges. Several individuals showed small home ranges when compared to other cetacean species. An overlap of home ranges of different sizes and shapes were observed for Guiana dolphins with large range movements. Centres of activity were concentrated in the main entrance of the Cananéia estuary. This was a first attempt to understand the way S. guianensis uses the Cananéia estuary and such data are essential for conservation and management purposes.