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 \({L(c)= \{F(x) \ge c \}}\), 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.
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Notes
This requires that the partial derivatives of the regression function \(r\) are \(k\)-Hölderian with a constant \(C\) (for further details see Definition 1 in Kohler et al. 2009).
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Acknowledgments
The authors thank the two anonymous reviewers and an associated editor for their useful comments and suggestions. This work has been partially supported by the French research national agency (ANR) under the references ANR-08BLAN-0314-01 and ANR 2011 BS01 010 01 projet Calibration. The authors thank Yannick Baraud, Roland Diel, Christine Tuleau-Malo and Patricia Reynaud-Bouret for fruitful discussions.
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Di Bernardino, E., Laloë, T. & Servien, R. Estimating covariate functions associated to multivariate risks: a level set approach. Metrika 78, 497–526 (2015). https://doi.org/10.1007/s00184-014-0498-4
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DOI: https://doi.org/10.1007/s00184-014-0498-4