🤖 AI Summary
This study addresses the inefficiency of traditional quantile estimation under light-tailed, heavy-tailed, or asymmetric distributions and its difficulty in smoothly bridging central and tail regions. The authors propose a unified interpolation-based quantile estimation framework that incorporates quadratic, Huber, or Tukey bisquare regularization into the check loss function, enabling continuous control of the effective quantile level via an interpolation parameter. They derive, for the first time, a closed-form parametrization of the effective quantile level under quadratic interpolation and establish a complete asymptotic theory, revealing the dependence of estimation efficiency on distributional shape. Theoretical and simulation results demonstrate that the proposed method reduces asymptotic variance by up to 36% under light-tailed distributions and by up to 57% under heavy-tailed or asymmetric distributions. Empirical analysis of daily log-returns confirms its superior performance in tail risk estimation.
📝 Abstract
This paper introduces a unified family of interpolated quantile estimators obtained by augmenting the check loss with quadratic, Huber, or Tukey's bisquare regularization. The estimators are indexed by the quantile level $τ$ and an interpolation parameter $h$. They reduce to the classical empirical quantile when $h=0$, while increasing $h$ continuously shifts the effective probability level toward the center of the distribution. A complete asymptotic theory is developed. For the quadratic interpolation, the effective quantile level is characterized by an interpolation equation yielding a closed-form parametrization of neighboring quantiles. Asymptotic normality is established for all three interpolated estimators via M-estimation, and a decomposition of the asymptotic variance explains how efficiency depends on the underlying distribution. Numerical experiments show that the quadratic interpolated estimator can reduce asymptotic variance by up to 36\% for light-tailed distributions and up to 57\% for heavy-tailed or asymmetric distributions for suitable interpolation strength. The framework is extended to linear quantile regression, where Monte Carlo experiments show that Huber interpolation is beneficial only in a narrow neighborhood of the median, while ordinary quantile regression remains preferable elsewhere. An application to daily log-returns illustrates the practical relevance of the proposed methodology for tail estimation under heavy tails and asymmetry.