Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory

Laloë

Servien

2014

Metrika

Self Cite

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.

“…From Assumption H (Section 2) and Proposition 2.1 in Di Bernardino et al [15], it follows that there exists a γ > 0 such that, if 2 ε n ≤ γ then …”

Section: Ali-mikhail-haq Copulamentioning

confidence: 99%

Section: Covariate-conditional-tail-expectation Consistencymentioning

confidence: 99%

Section: Independent Copulamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Laloë

Servien

2014

Metrika

Self Cite

“…A real case in environmental framework is also analyzed. The performances of our new estimator are compared to the ones of the level sets-based estimator, previously proposed in Di Bernardino et al (2011). …”

mentioning

confidence: 99%

Estimation of multivariate conditional-tail-expectation using Kendall's process

Journal of Nonparametric Statistics

Prieur

2014

Self Cite

This paper deals with the problem of estimating the Multivariate version of the Conditional-TailExpectation, proposed by Cousin and Di Bernardino (2012). We propose a new non-parametric estimator for this multivariate risk-measure, which is essentially based on the Kendall's process (see Genest and Rivest, 1993). Using the Central Limit Theorem for the Kendall's process, proved by Barbe et al. (1996), we provide a functional Central Limit Theorem for our estimator. We illustrate the practical properties of our estimator on simulations. A real case in environmental framework is also analyzed. The performances of our new estimator are compared to the ones of the level sets-based estimator, previously proposed in Di Bernardino et al. (2011).

Estimation of extreme quantiles conditioning on multivariate critical layers

Palacios‐Rodríguez

2016

Environmetrics

Let Ti:=[Xi|X∈∂L(α)], for i = 1,…,d, where X = (X1,…,Xd) is a risk vector and ∂L(α) is the associated multivariate critical layer at level α∈(0,1). The aim of this work is to propose a non‐parametric extreme estimation procedure for the (1 − pn)‐quantile of Ti for a fixed α and when pn→0, as the sample size n→+∞. An extrapolation method is developed under the Archimedean copula assumption for the dependence structure of X and the von Mises condition for marginal Xi. The main result is the central limit theorem for our estimator for p = pn→0, when n tends towards infinity. A set of simulations illustrates the finite‐sample performance of the proposed estimator. We finally illustrate how the proposed estimation procedure can help in the evaluation of extreme multivariate hydrological risks. Copyright © 2016 John Wiley & Sons, Ltd.