Estimating covariate functions associated to multivariate risks: a level set approach

Abstract

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) ≥ c}, with c ∈ (0, 1), of an unknown distribution function F on 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 Lp-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. Finally, a real environmental application of our risk measure is provided.

Exact asymptotic limit for kernel estimation of regression level sets

“…This denition of t can be of interest for practical applications, for example, when an insurance company wishes to focus on cost levels for unlikely risk factor values. In this case, t denes a level such that the set L(t) of risk values for which the cost is greater than this level is of probability P(L(t)) = p. In this spirit, we can mention the hydrology application presented in Di Bernardino et al (2015). For this problem, we need a more restrictive bandwidth assumption (which implies assumption B2):…”