Tail dimension reduction for extreme quantile estimation
Given covariates \((Y,X) \in \mathbb{R} \times \mathbb{R}^p\), a problem of quantile estimation \(Q(\alpha|X)=\inf\{y:\mathbb{P}(Y>y|X)\leq \alpha\}\) is considered under the assumption \(\alpha \to 0\), i.e., the problem of quantile estimation for the tail of the conditional distribution. In this case, the support of \(X\) may be only sparsely populated by data points if the dimension \(p\) is large. In the article, it is proposed to reduce the dimension by the replacement of \(X\) by \(B^TX\), where \(B\) is a \((p\times q)\)-matrix and \(q < p\). Conditions for the equivalence of \(Q(\alpha|X)\) and \(Q(\alpha|B^TX)\), the determination of an appropriate matrix \(B\) and a quantile estimator are presented. The results are illustrated on simulated data and on a real dataset.
scientific article; zbMATH DE number 6860552
scientific article