Near Quantile Regression Methods

• Near quantile regression is a modified approach that replaces the non-differentiable check loss with a smooth ℓₚ-quantile loss (p > 1) to retain robust estimation properties.
• It preserves the quantile identification property and asymptotic normality while enabling gradient-based optimization and adaptive penalization in high-dimensional settings.
• The method improves computational efficiency and variance estimation in challenging scenarios, making it ideal for heavy-tailed and large-scale data applications.

Near quantile regression denotes a set of methodologies that approximate or regularize classical quantile regression by modifying or smoothing the loss function, thereby retaining or improving robustness and interpretability while enabling efficient computation and refined inference in high-dimensional, heavy-tailed, or large-scale data environments. These methods typically interpolate between the canonical $\ell_1$-based quantile loss and smoother, differentiable alternatives, maintaining the quantile identification property while benefiting from the numerical and analytic advantages of smooth objective functions. The concept has particular relevance for model selection, high-dimensional estimation, variance estimation, and uniform inference.

1. Theoretical Rationale for Near Quantile Regression

Classical quantile regression, based on the “check” loss function $\rho_{\tau}(s) = (\tau - 1\{ s < 0 \})|s|$, is robust and invariant to heteroscedasticity and heavy tails. However, the non-differentiability of the absolute value at zero causes multiple challenges:

• Analytic complications in establishing higher-order asymptotics or constructing refined uniform approximations to the sampling distribution (as with Bahadur representation, which yields only $O(n^{-1/4})$ error).
• Numerical inefficiency in high-dimensions or large-scale settings, where reliance on linear or interior-point programming becomes computationally expensive and memory-intensive.
• Difficulty in variance estimation, where classic approaches require careful nonparametric density estimation at the quantile, a procedure sensitive to bandwidth choices and data sparsity.

Near quantile regression addresses these challenges by modifying the loss function to obtain a surrogate that is smooth and differentiable, while remaining a close approximation—*in the sense of the induced estimator—to the classical quantile parameter. The family of $\ell_p$-quantile losses is a canonical example: $\eta_{\tau,p}(s) = |\tau - 1\{s < 0\}|\,|s|^p.$ For $1 < p \leq 2,$ as $p \to 1^+$, $\eta_{\tau,p}$ converges pointwise to the check loss but is smooth for $p > 1$.

2. Mathematical Formulation and Asymptotic Properties

Given the linear model $y_t = x_t^\top \beta_0 + u_t$ with the $\tau$th quantile of $u_t$ set to zero, the “near quantile regression” estimator (for fixed $p > 1$ close to 1) is defined as: $$\hat{\beta}_{T,p}(\tau) = \underset{\beta \in \mathbb{R}^p}{\arg\min} \frac{1}{T} \sum_{t=1}^T \left[ \eta_{\tau,p}(y_t - x_t^\top \beta) - \eta_{\tau,p}(y_t) \right],$$ where the term $\eta_{\tau,p}(y_t)$ acts as a centering constant to ensure identifiability.

Asymptotic normality is established under appropriate regularity conditions. The main result is that as $T \to \infty$ and $p \to 1^+$ jointly,

$$\sqrt{T} \left( \hat{\beta}_{T,p}(\tau) - \beta(\tau) \right) \stackrel{d}{\to} \mathcal{N}(0, \Sigma_0),$$

where $\Sigma_0 = \tau(1 - \tau) f(0)^{-2} D_0^{-1}$ and $D_0$ is the limiting design matrix. This matches the asymptotic distribution of the standard quantile regression estimator, validating the near quantile approach as statistically equivalent in large samples (Lin, 20 Oct 2025).

This smooth reformulation also enables a new approach to estimating the asymptotic covariance matrix that bypasses direct nonparametric density estimation at the quantile.

3. Smoothing, Computation, and High-Dimensional Considerations

The primary computational advantage of a near quantile (smoothed) objective is enabling the use of gradient-based optimization in high dimensions, in contrast to the need for LP, IP, or active set methods for non-smooth problems.

• Algorithmic Framework: The paper proposes a cyclic coordinate descent and augmented proximal gradient algorithm (“CCPA”), which leverages the differentiability of the $\ell_p$-quantile loss for $p>1$. Each coordinate is updated efficiently using a soft-thresholding operator to accommodate $\ell_1$ regularization for variable selection in high-dimensional settings.
• Scalability: The approach achieves high efficiency in large $T$, high $p$ problems and generalizes naturally to composite (multiple quantile level) objectives often used in robust estimation or simultaneous quantile inference.
• Oracle Properties: For penalized versions with adaptive $\ell_1$ penalties, the method enjoys theoretical guarantees akin to oracle model selection under standard conditions (Lin, 20 Oct 2025).

4. Statistical and Practical Implications

A key advantage of near quantile regression is the ability to perform robust estimation and inference in scenarios where traditional quantile regression faces obstacles:

• Asymptotic Covariance Estimation: The smoothing ensures that one can construct consistent estimators of the limiting variance without the instability associated with kernel density estimation at the quantile.
• Heavy-Tailed and Infinite-Variance Errors: In the context of composite $\ell_p$-quantile regression (CLpQR), with $p$ near 1, the estimator can be more efficient than least-squares or composite quantile regression (CQR) when error variance is infinite. In some settings, the efficiency gain can be arbitrarily large relative to classical estimators.
• No Ad-Hoc Smoothing: This approach produces smoothed objective functions “by design,” eliminating the need for separate kernel smoothing or local linear approximations for the quantile process.

5. Comparative Perspective: Relation to Other Smoothing Approaches

While other smoothing strategies for quantile regression exist (e.g., kernel smoothing, rearrangement or enforcing monotonicity, spline quantile regression (Li et al., 7 Jan 2025)), near quantile regression as formalized via the $\ell_p$-quantile loss provides a principled and integrated route to achieve both smoothing and computational tractability:

Method	Smoothness	Directly targets quantile	Algorithmic tractability
Check loss (p=1)	Non-smooth	Yes	Linear/interior point only
$\ell_p$ loss ($p\to1^+$)	Smooth	Yes, as $p \to 1^+$	Gradient/proximal methods
Kernel smoothing	Smooth	Approximate	Bandwidth selection required
Spline QR (Li et al., 7 Jan 2025)	Smooth over $\tau$	Yes	LP/interior point or gradient
Composite Lp-QR (Lin, 20 Oct 2025)	Smooth	Yes, $\forall\, p>1$	CCPA, allows penalized high-D

This demonstrates that near quantile regression, via the $\ell_p$ loss with $p \to 1^+$, offers a unique blend of robustness, computation, and theoretical tractability, especially suited for modern statistical environments.

6. Implementation and Limitations

Implementation centers around the cyclic coordinate descent–proximal gradient hybrid for high-dimensional penalized regression, utilizing the differentiable structure of the $\ell_p$-quantile loss. Notably, the algorithm automatically specializes to the standard check-loss case if $p = 1$, ensuring full compatibility. The main steps:

1. Stack intercept(s) and coefficients into a single vector $\alpha$.
2. Iterate over coordinates, minimize the one-dimensional smooth composite loss plus penalty.
3. Apply soft-thresholding as the proximal operator for the penalty.
4. Stop according to a convergence criterion.

Potential limitations:

• Selecting $p$ close to 1 maintains fidelity to the quantile target but as $p$ rises convergence to quantile regression properties can, in principle, be less sharp.
• In implementation, very small $p-1$ may require careful numerical tuning to balance smoothness and approximation.

7. Applications and Future Directions

Near quantile regression extends classical quantile methodology to data-rich, high-dimensional, or heavy-tailed error settings, making it particularly relevant for genomics, finance, signal processing, and any context where computational efficiency and robust inference for conditional quantiles are required. Future directions suggested in (Lin, 20 Oct 2025) include:

• Refining the method as $p \to 1$ for local inference or nonparametric extensions.
• Further developing shrinkage techniques that exploit the smoothness of the loss for enhanced support recovery.
• Exploring nonparametric or semiparametric analogues.
• Investigating tailoring the degree of smoothness $p$ in a data-adaptive or theoretically optimal manner, especially in composite quantile regression frameworks.

Summary

Near quantile regression, as formalized in (Lin, 20 Oct 2025), is a theoretically-justified, smooth, and computationally efficient alternative to classical quantile regression. By employing the $\ell_p$-quantile loss for $p$ arbitrarily close to 1, the methodology produces estimators that are statistically (asymptotically) equivalent to the classical case, simplifies variance estimation, and admits scalable and robust penalties for high-dimensional model selection. This framework advances both the theory and efficient computation of quantile regression in modern, complex data contexts.