The aim of this paper is to study the asymptotic behavior of a particular multivariate risk measure, the Covariate-Conditional-Tail-Expectation (CCTE), based on a multivariate statistical depth function. Depth functions have become increasingly powerful tools in nonparametric inference for multivariate data, as they measure a degree of centrality of a point with respect to a distribution. A multivariate risks scenario is then represented by a depth-based lower level set of the risk factors, meaning that we consider a non-compact setting. More precisely, given a multivariate depth function D associated to a fixed probability measure, we are interested in the lower level set based on D. First, we present a plug-in approach in order to estimate the depth-based level set. In a second part, we provide a consistent estimator of our CCTE for a general depth function with a rate of convergence, and we consider the particular case of the Mahalanobis depth. A simulation study complements the performances of our estimator.
翻译:本文的目的是研究基于多变量统计深度功能的某个特定多变量风险计量( Covoliate-Sotitual-Tail-Explectation)的无症状行为。 深度函数在多变量数据的非参数推论中变得日益强大, 因为它们测量了一个点相对于分布的中心度。 一种多变量风险假设情景由一组基于深度的较低风险因素代表, 这意味着我们考虑一个非兼容性设置。 更确切地说, 鉴于一个与固定概率计量相关的多变量深度函数D, 我们感兴趣的是基于 D 设定的较低水平。 首先, 我们提出了一个插插座法, 以估计基于深度的定值。 在第二部分, 我们为我们的 CCTE 提供了一个一致的测算器, 用于一个具有趋同率的一般深度函数, 我们考虑了马哈拉诺比深度的特殊情况。 模拟研究补充了我们测算器的性能 。