TY - JOUR

T1 - Nonparametric estimation of conditional marginal excess moments

AU - Goegebeur, Yuri

AU - Guillou, Armelle

AU - Ho, Nguyen Khanh Le

AU - Qin, Jing

N1 - Funding Information:
The authors sincerely thank the editor, associate editor and the referee for their helpful comments and suggestions that led to considerable improvement of the paper. The research of Armelle Guillou was supported by the French National Research Agency under the grant ANR-19-CE40-0013-01/ExtremReg project and an International Emerging Action (IEA-00179). Computation/simulation for the work described in this paper was supported by the DeIC National HPC Centre, SDU.
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2023/1

Y1 - 2023/1

N2 - Several risk measures have been proposed in the literature, among them the marginal mean excess, defined as MMEp=E[{Y(1)−Q1(1−p)}+|Y(2)>Q2(1−p)], provided E|Y(1)|<∞, where (Y(1),Y(2)) denotes a pair of risk factors, y+≔max(0,y), Qj the quantile function of Y(j),j∈{1,2}, and p∈(0,1). In this paper we consider a generalization of this measure, where the random variables of main interest (Y(1),Y(2)) are observed together with a random covariate X∈Rd, and where the Y(1) excess is also power transformed. This leads to the concept of conditional marginal excess moment for which an estimator is proposed allowing extrapolation outside the data range. The main asymptotic properties of this estimator have been established, using empirical processes arguments combined with the multivariate extreme value theory. The finite sample behavior of the estimator is evaluated by a simulation experiment. We apply also our method on a vehicle insurance customer dataset.

AB - Several risk measures have been proposed in the literature, among them the marginal mean excess, defined as MMEp=E[{Y(1)−Q1(1−p)}+|Y(2)>Q2(1−p)], provided E|Y(1)|<∞, where (Y(1),Y(2)) denotes a pair of risk factors, y+≔max(0,y), Qj the quantile function of Y(j),j∈{1,2}, and p∈(0,1). In this paper we consider a generalization of this measure, where the random variables of main interest (Y(1),Y(2)) are observed together with a random covariate X∈Rd, and where the Y(1) excess is also power transformed. This leads to the concept of conditional marginal excess moment for which an estimator is proposed allowing extrapolation outside the data range. The main asymptotic properties of this estimator have been established, using empirical processes arguments combined with the multivariate extreme value theory. The finite sample behavior of the estimator is evaluated by a simulation experiment. We apply also our method on a vehicle insurance customer dataset.

KW - Empirical process

KW - Marginal mean excess

KW - Pareto-type distribution

KW - Tail dependence

U2 - 10.1016/j.jmva.2022.105121

DO - 10.1016/j.jmva.2022.105121

M3 - Journal article

AN - SCOPUS:85141286763

SN - 0047-259X

VL - 193

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

M1 - 105121

ER -