Nonparametric estimation of conditional marginal excess moments

Yuri Goegebeur, Armelle Guillou*, Nguyen Khanh Le Ho, Jing Qin

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review


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.

Original languageEnglish
Article number105121
JournalJournal of Multivariate Analysis
Publication statusPublished - Jan 2023


  • Empirical process
  • Marginal mean excess
  • Pareto-type distribution
  • Tail dependence


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