This paper deals with the problem of estimating the level sets L(c) = {F(x) ≥ c}, with c ∈ (0,1), of an unknown distribution function F on ℝ+ 2. A plug-in approach is followed. That is, given a consistent estimator F n of F, we estimate L(c) by L n (c) = {F n (x) ≥ c}. In our setting, non-compactness property is a priori required for the level sets to estimate. We state consistency results with respect to the Hausdorff distance and the volume of the symmetric difference. Our results are motivated by applications in multivariate risk theory. In particular we propose a new bivariate version of the conditional tail expectation by conditioning the two-dimensional random vector to be in the level set L(c). We also present simulated and real examples which illustrate our theoretical results.

The aim of this paper is to study the behavior of a covariate function in a multivariate risks scenario. The first part of this paper deals with the problem of estimating the \(c\) -upper level sets \({L(c)= \{F(x) \ge c \}}\) , with \(c \in (0,1)\) , of an unknown distribution function \(F\) on \(\mathbb {R}^d_+\) . A plug-in approach is followed. We state consistency results with respect to the volume of the symmetric difference. In the second part, we obtain the \(L_p\) -consistency, with a convergence rate, for the regression function estimate on these level sets \(L(c)\) . We also consider a new multivariate risk measure: the Covariate-Conditional-Tail-Expectation. We provide a consistent estimator for this measure with a convergence rate. We propose a consistent estimate when the regression cannot be estimated on the whole data set. Then, we investigate the effects of scaling data on our consistency results. All these results are proven in a non-compact setting. A complete simulation study is detailed and a comparison with parametric and semi-parametric approaches is provided. Finally, a real environmental application of our risk measure is provided.

Highly resolved and accurate daily precipitation data are required for impact models to perform adequately and correctly measure the impacts of high-risk events. In order to produce such data, bias correction is often needed. Most of those statistical methods correct the probability distributions of daily precipitation by modeling them with either empirical or parametric distributions. A recent semi-parametric model based on a penalized Berk–Jones (BJ) statistical test, which allows for automatic and personalized splicing of parametric and non-parametric distributions, has been developed. This method, called the Stitch-BJ model, was found to be able to model daily precipitation correctly and showed interesting potential in a bias correction setting. In the present study, we will consolidate these results by taking into account the seasonal properties of daily precipitation in an out-of-sample context and by considering dry days probabilities in our methodology. We evaluate the performance of the Stitch-BJ method in this seasonal bias correction setting against more classical models such as the Gamma, Exponentiated Weibull (ExpW), Extended Generalized Pareto (EGP) or empirical distributions. Results show that a seasonal separation of data is necessary in order to account for intra-annual non-stationarity. Moreover, the Stitch-BJ distribution was able to consistently perform as well as or better than all the other considered models over the validation set, including the empirical distribution, which is often used due to its robustness. Finally, while methods for correcting dry day probabilities can be easily applied, their relevance can be discussed as temporal and spatial correlations are often neglected.