鲁棒插值分位数估计量：渐近理论与效率
Robust Interpolated Quantile Estimators: Asymptotic Theory and Efficiency
AI总结：
本文提出含三类正则项的插值分位数估计量，建立其渐近理论，数值实验显示二次插值估计量可降低渐近方差，扩展至线性分位数回归后验证了其在尾部估计的实用价值。
AI中文摘要：
本文提出一类统一的插值分位数估计量，通过在检查损失（check loss）中加入二次、Huber或Tukey双平方（bisquare）正则项得到。这类估计量由分位数水平τ和插值参数h索引；当h=0时退化为经典经验分位数，增大h会使有效概率水平向分布中心连续偏移。本文建立了完整的渐近理论：对于二次插值，有效分位数水平由插值方程刻画，可得到相邻分位数的闭式参数化；通过M估计法证明了三类插值估计量的渐近正态性，渐近方差的分解解释了效率如何依赖于基础分布。数值实验表明，对于轻尾分布，二次插值估计量在合适的插值强度下可降低渐近方差达36%；对于重尾或非对称分布，该降幅可达57%。该框架被扩展至线性分位数回归：蒙特卡洛实验显示，Huber插值仅在中位数附近的狭窄区间内有益，普通分位数回归在其他场景仍更优。将其应用于日对数收益率，验证了所提方法在重尾与非对称场景下尾部估计的实用价值。
英文摘要：
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.
