Estimation of Extreme Conditional Quantiles Through Power Transformation

Abstract

The estimation of extreme conditional quantiles is an important issue in numerous disciplines. Quantile regression (QR) provides a natural way to capture the covariate effects at different tails of the response distribution. However, without any distributional assumptions, estimation from conventional QR is often unstable at the tails, especially for heavy-tailed distributions due to data sparsity. In this article, we develop a new three-stage estimation procedure that integrates QR and extreme value theory by estimating intermediate conditional quantiles using QR and extrapolating these estimates to tails based on extreme value theory. Using the power-transformed QR, the proposed method allows more flexibility than existing methods that rely on the linearity of quantiles on the original scale, while extending the applicability of parametric models to borrow information across covariates without resorting to nonparametric smoothing. In addition, we propose a test procedure to assess the commonality of extreme value index, which could be useful for obtaining more efficient estimation by sharing information across covariates. We establish the asymptotic properties of the proposed method and demonstrate its value through simulation study and the analysis of a medical cost data. Supplementary materials for this article are available online.

KEY WORDS:

• Box–Cox power transformation
• Extreme value
• Heavy-tailed distribution
• High quantile
• Quantile regression

Acknowledgments

The research is partially supported by the NSF Award DMS-1007420, NSF CAREER Award DMS-1149355, and NNSFC grants 11171074 and 71232008.