Aggregation of log-linear risks

In this paper we work in the framework of a k-dimensional vector of log-linear risks. Under weak conditions on the marginal tails and the dependence structure of a vector of positive risks, we derive the asymptotic tail behaviour of the aggregated risk and present an application concerning log-normal risks with stochastic volatility

Aggregation of randomly weighted large risks

Asymptotic tail probabilities for linear combinations of randomly weighted order statistics are approximated under various assumptions. One key assumption is the asymptotic independence for all risks. Therefore, it is not surprising that the maxima represents the most influential factor when one investigates the tail behaviour of our considered risk aggregation, which, for example, can be found in the reinsurance market. This extreme behaviour confirms the ‘one big jump’ property that has been vastly discussed in the existing literature in various forms whenever asymptotic independence is present. An illustration of our results together with a specific application are explored under the assumption that the underlying risks follow the multivariate log-normal distribution

Tail asymptotics of randomly weighted large risks

Tail asymptotic probabilities for linear combinations of randomly weighted order statistics are approximated under various assumptions. One key assumption is the asymptotic independence for all risks, and thus, it is not surprising that the maxima represents the most influential factor when one investigates the tail behaviour of our considered risk aggregation, which for example, can be found in the reinsurance market. This extreme behaviour confirms the “one big jump” property that has been vastly discussed in the existing literature in various forms whenever the asymptotic independence is present. An illustration of our results together with a specific application are explored under the assumption that the underlying risks follow the multivariate Log-normal distribution. Keywords and phrases: Davis-Resnick tail property; Extreme value distribution; Max-domain of attraction; Mitra-Resnick model; Risk aggregatio

Aggregation for Gaussian regression

This paper studies statistical aggregation procedures in the regression setting. A motivating factor is the existence of many different methods of estimation, leading to possibly competing estimators. We consider here three different types of aggregation: model selection (MS) aggregation, convex (C) aggregation and linear (L) aggregation. The objective of (MS) is to select the optimal single estimator from the list; that of (C) is to select the optimal convex combination of the given estimators; and that of (L) is to select the optimal linear combination of the given estimators. We are interested in evaluating the rates of convergence of the excess risks of the estimators obtained by these procedures. Our approach is motivated by recently published minimax results [Nemirovski, A. (2000). Topics in non-parametric statistics. Lectures on Probability Theory and Statistics (Saint-Flour, 1998). Lecture Notes in Math. 1738 85--277. Springer, Berlin; Tsybakov, A. B. (2003). Optimal rates of aggregation. Learning Theory and Kernel Machines. Lecture Notes in Artificial Intelligence 2777 303--313. Springer, Heidelberg]. There exist competing aggregation procedures achieving optimal convergence rates for each of the (MS), (C) and (L) cases separately. Since these procedures are not directly comparable with each other, we suggest an alternative solution. We prove that all three optimal rates, as well as those for the newly introduced (S) aggregation (subset selection), are nearly achieved via a single ``universal'' aggregation procedure. The procedure consists of mixing the initial estimators with weights obtained by penalized least squares. Two different penalties are considered: one of them is of the BIC type, the second one is a data-dependent $\ell_1$-type penalty.Comment: Published in at http://dx.doi.org/10.1214/009053606000001587 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Tail asymptotics of randomly weighted large risks

In this paper we are concerned with a sample of asymptotically independent risks. Tail asymptotic probabilities for linear combinations of randomly weighted order statistics are approximated under various assumptions, where the individual tail behaviour has a crucial role. An application is provided for Log-Normal risks

Multivariate space-time modelling of multiple air pollutants and their health effects accounting for exposure uncertainty

The long-term health effects of air pollution are often estimated using a spatio-temporal ecological areal unit study, but this design leads to the following statistical challenges: (1) how to estimate spatially representative pollution concentrations for each areal unit; (2) how to allow for the uncertainty in these estimated concentrations when estimating their health effects; and (3) how to simultaneously estimate the joint effects of multiple correlated pollutants. This article proposes a novel 2-stage Bayesian hierarchical model for addressing these 3 challenges, with inference based on Markov chain Monte Carlo simulation. The first stage is a multivariate spatio-temporal fusion model for predicting areal level average concentrations of multiple pollutants from both monitored and modelled pollution data. The second stage is a spatio-temporal model for estimating the health impact of multiple correlated pollutants simultaneously, which accounts for the uncertainty in the estimated pollution concentrations. The novel methodology is motivated by a new study of the impact of both particulate matter and nitrogen dioxide concentrations on respiratory hospital admissions in Scotland between 2007 and 2011, and the results suggest that both pollutants exhibit substantial and independent health effects