Robust Interpolated Quantile Estimators: Asymptotic Theory and Efficiency
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.