Smoothed Quantile Estimation via Interpolation to the Mean

Abstract

This paper introduces a unified family of smoothed quantile estimators that continuously interpolate between classical empirical quantiles and the sample mean. The estimators q(z, h) are defined as minimizers of a regularized objective function depending on two parameters: a smoothing parameter h $\ge$ 0 and a location parameter z $\in$ R. When h = 0 and z $\in$ (-1, 1), the estimator reduces to the empirical quantile of order $\tau$ = (1z)/2; as h $\rightarrow$ $\infty$, it converges to the sample mean for any fixed z. We establish consistency, asymptotic normality, and an explicit variance expression characterizing the efficiency-robustness trade-off induced by h. A key geometric insight shows that for each fixed quantile level $\tau$ , the admissible parameter pairs (z, h) lie on a straight line in the parameter space, along which the population quantile remains constant while asymptotic efficiency varies. The analysis reveals two regimes: under light-tailed distributions (e.g., Gaussian), smoothing yields a monotonic but asymptotic variance reduction with no finite optimum; under heavy-tailed distributions (e.g., Laplace), a finite smoothing level h * ($\tau$ ) > 0 achieves strict efficiency improvement over the classical empirical quantile. Numerical illustrations confirm these theoretical predictions and highlight how smoothing balances robustness and efficiency across quantile levels.