Improved kernel estimation of copulas: Weak convergence and goodness-of-fit testing

We reconsider the existing kernel estimators for a copula function, as proposed in Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445--464], Fermanian, Radulovi\v{c} and Wegkamp [Bernoulli 10 (2004) 847--860] and Chen and Huang [Canad. J. Statist. 35 (2007) 265--282]. All of these estimators have as a drawback that they can suffer from a corner bias problem. A way to deal with this is to impose rather stringent conditions on the copula, outruling as such many classical families of copulas. In this paper, we propose improved estimators that take care of the typical corner bias problem. For Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445--464] and Chen and Huang [Canad. J. Statist. 35 (2007) 265--282], the improvement involves shrinking the bandwidth with an appropriate functional factor; for Fermanian, Radulovi\v{c} and Wegkamp [Bernoulli 10 (2004) 847--860], this is done by using a transformation. The theoretical contribution of the paper is a weak convergence result for the three improved estimators under conditions that are met for most copula families. We also discuss the choice of bandwidth parameters, theoretically and practically, and illustrate the finite-sample behaviour of the estimators in a simulation study. The improved estimators are applied to goodness-of-fit testing for copulas.Comment: Published in at http://dx.doi.org/10.1214/08-AOS666 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Evaluation of pre/post-fire differenced spectral indices for assessing burn severity in a Mediterranean environment with landsat thematic mapper

In this study several pre/post-fire differenced spectral indices for assessing burn severity in a Mediterranean environment are evaluated. GeoCBI (Geo Composite Burn Index) field data of burn severity were correlated with remotely sensed measures, based on the NBR (Normalized Burn Ratio), the NDMI (Normalized Difference Moisture Index) and the NDVI (Normalized Difference Vegetation Index). In addition, the strength of the correlation was evaluated for specific fuel types and the influence of the regression model type is pointed out. The NBR was the best remotely sensed index for assessing burn severity, followed by the NDMI and the NDVI. For this case study of the 2007 Peloponnese fires, results show that the GeoCBI-dNBR (differenced NBR) approach yields a moderate-high R(2) = 0.65. Absolute indices outperformed their relative equivalents, which accounted for pre-fire vegetation state. The GeoCBI-dNBR relationship was stronger for forested ecotypes than for shrub lands. The relationship between the field data and the dNBR and dNDMI (differenced NDMI) was nonlinear, while the GeoCBI-dNDVI (differenced NDVI) relationship appeared linear

Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation

In the renewal risk model, we study the asymptotic behavior of the expected time-integrated negative part of the process. This risk measure has been introduced by Loisel (2005). Both heavy-tailed and light-tailed claim amount distributions are investigated. The time horizon may be finite or infinite. We apply the results to an optimal allocation problem with two lines of business of an insurance company. The asymptotic behavior of the two optimal initial reserves are computed.Ruin theory; heavy-tailed and light-tailed claim size distribution; risk measure; optimal reserve allocation