Estimation of extreme conditional quantiles under a general tail-first-order condition
The authors consider the problem of the estimation of an extreme conditional quantile. Let \(Y\) be a real-valued random variable and \(X\) be an explanatory random vector of dimension \(p \in \mathbb{N} \setminus \left\lbrace 0 \right\rbrace \). The conditional quantile of level \(\alpha \in [0,1]\) of \(Y\) given \(\left\lbrace X=x_{0}\right\rbrace \) is \(Q\left(\alpha\mid x_{0} \right) := \inf \left\lbrace y \in \mathbb{R}; \, S\left(y\mid x_{0} \right) \le \alpha \right\rbrace \), where \(S\left( \cdot\mid x_{0} \right) := \mathbb{P} \left(Y>\cdot\mid X=x_{0} \right) \) is the conditional survival function. Given \(n\) independent copies \(\left(X_{1},Y_{1}\right),\dots,\left(X_{n},Y_{n}\right) \) of \(\left(X,Y\right) \), they focus on the so-called indirect estimation method: starting from a suitable estimator \(\hat{S}_{n}\left(\cdot\mid x_{0} \right)\) of \(S\left(\cdot\mid x_{0} \right)\), the associated estimator of \(Q\left(\alpha\mid x_{0} \right)\) is given by \(\hat{Q}_{n}\left(\alpha\mid x_{0} \right) := \inf \left\lbrace y \in \mathbb{R}; \, \hat{S}_{n}\left(y\mid x_{0} \right) \le \alpha \right\rbrace\).
scientific article; zbMATH DE number 7223266
scientific article