Smoothed Quantile Estimation via Interpolation to the Mean
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915586912550912 |
|---|---|
| author | Maanan, Saïd Dermoune, Azzouz Ghini, Ahmed El |
| author_facet | Maanan, Saïd Dermoune, Azzouz Ghini, Ahmed El |
| contents | 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 $τ$ = (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 $τ$ , 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 * ($τ$ ) > 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26447 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Smoothed Quantile Estimation via Interpolation to the Mean Maanan, Saïd Dermoune, Azzouz Ghini, Ahmed El Methodology 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 $τ$ = (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 $τ$ , 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 * ($τ$ ) > 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. |
| title | Smoothed Quantile Estimation via Interpolation to the Mean |
| topic | Methodology |
| url | https://arxiv.org/abs/2510.26447 |