Tail dimension reduction for extreme quantile estimation

Abstract

In a regression context where a response variable Y ∈ R is recorded with a covariate X ∈ R p , two situations can occur simultaneously: (a) we are interested in the tail of the conditional distribution and not on the central part of the distribution and (b) the number p of regressors is large. To our knowledge, these two situations have only been considered separately in the literature. The aim of this paper is to propose a new dimension reduction approach adapted to the tail of the distribution in order to propose an efficient conditional extreme quantile estimator when the dimension p is large. The results are illustrated on simulated data and on a real dataset.

Extreme Quantile Estimation Based on the Tail Single-index Model

“…In Case 2, the tail single-index model assumption in (2.1) is satisfied. Case 3 is a multi-index model that depends on two indices and satisfies the TDR space assumption in Gardes (2018). As τ → 1, the quantile of Y depends on x approximately through the single index…”

“…There exist some work that integrates the single-index model and quantile regression; see Wu et al (2010), Zhu et al (2012), Kong and Xia (2012), Zhong et al (2016), among others. To our best knowledge, there is only one work (Gardes, 2018) that discussed the estimation of extreme conditional quantiles for single-index and multi-index models. Gardes (2018) proposed a new dimension reduc-tion approach and a conditional extremal quantile estimator by considering the tail dimension reduction subspace.…”

“…To our best knowledge, there is only one work (Gardes, 2018) that discussed the estimation of extreme conditional quantiles for single-index and multi-index models. Gardes (2018) proposed a new dimension reduc-tion approach and a conditional extremal quantile estimator by considering the tail dimension reduction subspace. However, this method is computationally complex, and the paper did not formally establish the theoretical properties of the estimator when the index parameters are unknown and have to be estimated through data.…”

“…Secondly, the proposed tail single-index model not only provides more flexibility than parametric models, but also leads to a simple approach for estimating the index parameters with a √ n-convergence rate so that this estimation does not affect the asymptotic properties of the ultimate extreme quantile estimation. In contrast, the index estimation method in Gardes (2018) is more complicated and numerically less stable, and its theoretical properties and impacts on the extreme quantile estimation were not formally studied. Thirdly, instead of indirectly estimating conditional quantiles through inverting the conditional cumulative distribution function as in Gardes (2018), our procedure is based on the direct estimation of conditional quantiles in all three steps.…”

“…In contrast, the index estimation method in Gardes (2018) is more complicated and numerically less stable, and its theoretical properties and impacts on the extreme quantile estimation were not formally studied. Thirdly, instead of indirectly estimating conditional quantiles through inverting the conditional cumulative distribution function as in Gardes (2018), our procedure is based on the direct estimation of conditional quantiles in all three steps. We show that this coherence helps reduce errors from different layers of the modelling and ameliorates the tuning parameter selection, subsequently leading to numerically more accurate estimations.…”