Anne-Laure Fougères

Forecasts and their evaluation are major tasks in statistics. In real applications, forecasts often take the form of a dynamic process evolving over time and this sequential point of view must be taken into account. A strategy for forecast evaluation is calibration theory based on the Probability Integral Transform. The idea is to check the conformity between the forecast and the observation. Here, ideal forecasts are characterized by conditional calibration and we present some new tests based on regression trees.

Probabilistic forecast plays a major role in many applications where forecast is needed together with an assessment of its uncertainty. Verification of probabilistic forecast has become increasingly important and mostly relies on two sets of tools: scoring rules and calibration diagnostics. Proper scoring rules assign forecasts numerical scores such that the correct forecast achieves a minimal expected score. Calibration theory aims at verifying that the observations and the forecasts are consistent. In practice, using a probabilistic forecast commonly involves a sequential decision making process where the environment evolves over time. In this article, we propose a mathematical framework for dynamic probabilistic forecast. The forecasts take therein the form of stochastic processes adapted to a filtration that encodes the available information. Under minimal assumptions, we show that proper scoring rules can still be used in this dynamic framework to discriminate the ideal forecast-more precisely, we prove that the long term average score is close to minimum if and only if the forecasts are close to ideal. Some connections are also done in terms of Wasserstein distance, and links are given between scoring rules and reproducing kernels.

The modeling of insurance risks has received an increasing amount of attention because of solvency capital requirements. The ruin probability has become a standard risk measure to assess regulatory capital. In this paper we focus on discrete-time models for the finite time horizon. Several results are available in the literature to calibrate the ruin probability by means of the sum of the tail probabilities of individual claim amounts. The aim of this work is to obtain asymptotics for such probabilities under multivariate regular variation and, more precisely, to derive them from extensions of Breiman's theorem. We thus present new situations where the ruin probability admits computable equivalents. We also derive asymptotics for the value at risk.

The modeling of insurance risks has received an increasing amount of attention because of solvency capital requirements. The ruin probability has become a standard risk measure to assess regulatory capital. In this paper we focus on discrete-time models for the finite time horizon. Several results are available in the literature to calibrate the ruin probability by means of the sum of the tail probabilities of individual claim amounts. The aim of this work is to obtain asymptotics for such probabilities under multivariate regular variation and, more precisely, to derive them from extensions of Breiman's theorem. We thus present new situations where the ruin probability admits computable equivalents. We also derive asymptotics for the value at risk.

We investigate conditions for the existence of the limiting conditional distribution of a bivariate random vector when one component becomes large. We revisit the existing literature on the topic, and present some new sufficient conditions. We concentrate on the case where the conditioning variable belongs to the maximum domain of attraction of the Gumbel law, and we study geometric conditions on the joint distribution of the vector. We show that these conditions are of a local nature and imply asymptotic independence when both variables belong to the domain of attraction of an extreme value distribution. The new model we introduce can also be useful to simulate bivariate random vectors with a given limiting conditional distribution.