Abstract
Depth functions have become increasingly powerful tools in non-parametric inference for multivariate data as they measure a degree of centrality of a point with respect to a distribution. A multivariate risk scenario is then represented by a depth-based lower level set of the risk factors, meaning that we consider a non-compact setting. The aim of this paper is to study the asymptotic behavior of level sets of a general multivariate depth function and a particular multivariate risk measure, the Covariate-Conditional-Tail-Expectation (CCTE) based on a depth function. More precisely, given a probability measure P on and a depth function , we are interested in the α-lower level set . First, we present a plug-in approach in order to estimate , then we derive consistency of its estimator under some regularity conditions. In a second part, we provide a consistent estimator of the CCTE for a general depth function with a rate of convergence and we consider the particular case of Mahalanobis depth. Finally, a simulation study complements the performances of our estimator and an application on real data is presented.
Funding Statement
This work has been supported by the project ANR McLaren (ANR-20-CE23-0011).
Acknowledgments
The authors express their gratitude to two anonymous Referees and Associate Editor for their valuable comments on this article.
Citation
Sara Armaut. Roland Diel. Thomas Laloë. "Depth level set estimation and associated risk measures." Electron. J. Statist. 16 (2) 6584 - 6630, 2022. https://doi.org/10.1214/22-EJS2095
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