Abstract

This thesis takes place in extreme value statistics and agricultural insurance frameworks. For the first line of research, the extreme quantile of a response variable Y can often be linked to a vector of covariates X of dimension p. When p is large compared to the sample size n, the conditional distribution of Y given X becomes difficult to estimate, especially when dealing with extreme values. The first contribution of this thesis is to propose a new approach, called Extreme-PLS, for dimension reduction in conditional extreme values settings. This approach consists in reducing the dimension of X by maximizing the covariance between a linear combination of coordinates X and Y given large values of Y. We establish the asymptotic normality of the Extreme-PLS estimator under a single-index model. The second contribution provides a Bayesian extension to the Extreme-PLS method to address data scarcity problems in distribution tails. This approach allows to identify the direction of dimension reduction by introducing a prior information on it. It provides a Bayesian framework for computing the posterior distribution of the direction, where the likelihood function is obtained from a von Mises-Fisher distribution adapted to hyperballs. Three prior distributions are considered: conjugate, hierarchical and sparse priors. Finally, the performance of both approaches is evaluated on simulated data, and an application on French farm income data is provided as an illustration.

Regarding the second line of research, climate disruption and market deregulation have increased and impacted agricultural production and income. Farmers' incomes are faced with two main types of risk related to price and yield volatility. Protection against these risks fall within a good risk management and thus farmers' insurance coverage. The third contribution of this thesis concerns the study and modelling of the dependence structure between crop yield and price risks using copulas. We also use conditional copulas to take into account the effect of other covariates such as crop insurance purchase, claims and weather factors. The last contribution focuses on considering the natural hedge mechanism, i.e. the negative dependence between yields and prices, in a revenue insurance scheme. We analyse its effect on the value of the actuarially fair premium on an example of revenue insurance contract pricing. The results show that a natural hedge is likely to reduce insurance premiums in France.
All studies focus on French farm income data in the cereal (maize and wheat) and wine sectors.

Keywords: nonparametric statistic, extreme value theory, Bayesian statistic, insurance, natural disasters