# Smoothing quantile regressions

arXiv: https://arxiv.org/abs/1905.08535 · HTML: https://mlsanity.com/paper/1905.08535/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 May 2019.

## Built on by (most cited first, 16 of 16)

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- [Communication-Constrained Distributed Quantile Regression with Optimal Statistical Guarantees](https://mlsanity.com/paper/2110.13113/) (2021)
- [A Piecewise Lyapunov Analysis of Sub-quadratic SGD: Applications to Robust and Quantile Regression](https://mlsanity.com/paper/2504.08178/) (2025)
- [Distributed High-Dimensional Quantile Regression: Estimation Efficiency and Support Recovery](https://mlsanity.com/paper/2405.07552/) (2024)
- [Safe Collaborative Filtering](https://mlsanity.com/paper/2306.05292/) (2023)
- [ReModels: Quantile Regression Averaging models](https://mlsanity.com/paper/2405.11372/) (2024)
- [fastkqr: A Fast Algorithm for Kernel Quantile Regression](https://mlsanity.com/paper/2408.05393/) (2024)
- [Distributed Estimation and Inference for Semi-parametric Binary Response Models](https://mlsanity.com/paper/2210.08393/) (2022)
- [Private Optimal Inventory Policy Learning for Feature-based Newsvendor with Unknown Demand](https://mlsanity.com/paper/2404.15466/) (2024)
- [Fast networked data selection via distributed smoothed quantile estimation](https://mlsanity.com/paper/2406.01929/) (2024)
- [Smoothed Robust Phase Retrieval](https://mlsanity.com/paper/2409.01570/) (2024)
- [A Pathwise Coordinate Descent Algorithm for LASSO Penalized Quantile Regression](https://mlsanity.com/paper/2502.12363/) (2025)
- [Efficient Distributed Learning over Decentralized Networks with Convoluted Support Vector Machine](https://mlsanity.com/paper/2503.07563/) (2025)
- [Inertial Quadratic Majorization Minimization with Application to Kernel Regularized Learning](https://mlsanity.com/paper/2507.04247/) (2025)
- [ConquerNet: Convolution-Smoothed Quantile ReLU Neural Networks with Minimax Guarantees](https://mlsanity.com/paper/2605.06265/) (2026)

## Abstract (arXiv)

We propose to smooth the entire objective function, rather than only the check function, in a linear quantile regression context. Not only does the resulting smoothed quantile regression estimator yield a lower mean squared error and a more accurate Bahadur-Kiefer representation than the standard estimator, but it is also asymptotically differentiable. We exploit the latter to propose a quantile density estimator that does not suffer from the curse of dimensionality. This means estimating the conditional density function without worrying about the dimension of the covariate vector. It also allows for two-stage efficient quantile regression estimation. Our asymptotic theory holds uniformly with respect to the bandwidth and quantile level. Finally, we propose a rule of thumb for choosing the smoothing bandwidth that should approximate well the optimal bandwidth. Simulations confirm that our smoothed quantile regression estimator indeed performs very well in finite samples.