Online Inference for Censored Quantile Regression

• The paper introduces an online estimation framework that uses quantile calculus to derive integral equations for censored quantile regression.
• It employs a progressive localized minimization algorithm for real-time updates, ensuring numerical reliability and efficient convergence in high-dimensional settings.
• The method achieves uniform consistency and weak convergence, demonstrating practical advantages in clinical survival analysis over conventional approaches.

An online inference method for censored quantile regression provides a computational and inferential framework to estimate and continuously update quantile regression parameters in the presence of right-censored survival data, particularly as new data arrive in a streaming or sequential manner. The approach introduced by Huang in "Quantile calculus and censored regression" (1010.0514) is grounded in a general quantile calculus on the cumulative probability scale, leading to a unique estimation procedure for censored quantile regression. It resolves fundamental issues in prior approaches—such as grid dependence and algorithmic complexity—by casting the estimation as the solution to a set of integral equations and leveraging a progressive, localized minimization algorithm that is reliable and efficient for both inference and online updating.

1. Quantile Calculus and Integral Equations for Censored Quantile Regression

At the core of the methodology is the recognition that, under survival distributions with potential discontinuities or flat regions, the mapping between time and probability is not invertible everywhere. The quantile calculus is developed on the cumulative probability scale, introducing the "quantile equality fraction" ξ(τ) to handle mass at points of flatness or discontinuity.

For the one-sample case, the key relationship is:

$$\operatorname{Pr}\{T < Q(\tau)\} + \operatorname{Pr}\{T = Q(\tau)\} \xi(\tau) = \int_0^\tau \frac{\operatorname{Pr}\{T \geq Q(\nu)\}- \operatorname{Pr}\{T=Q(\nu)\}\xi(\nu)}{1-\nu} d\nu.$$

Extending to covariate-dependent (regression) settings, the conditional quantile is modeled as $Q_Z(\tau) = Z^\top \beta_0(\tau)$. In the presence of right censoring—where one observes $(X, \Delta) = (\min\{T, C\}, I(T \leq C))$—the estimating equation becomes:

$$E\left[ Z \Delta \left\{ I\left( X < Z^\top\beta(\tau) \right) + I\left( X = Z^\top\beta(\tau) \right) \eta_Z(\tau)\right\} \right] = E\left[ \int_0^\tau Z \left\{ I\left( X \geq Z^\top\beta(\nu) \right) - I\left( X = Z^\top\beta(\nu) \right) \eta_Z(\nu)\right\} \frac{d\nu}{1-\nu} \right],$$

where $\eta_Z(\tau)$ generalizes the quantile equality fraction to the regression setting. This integral equation serves as the backbone for both estimation and online inference.

2. Progressive Localized Minimization Algorithm (PLMIN)

The PLMIN algorithm constructs the full quantile coefficient process $\beta(\tau)$ for $\tau \in [0,1)$ as a cadlag function that evolves with $\tau$. At any quantile level, $\beta(\tau)$ defines a separating hyperplane in the covariate space, partitioning observations into above, below, and interpolated sets. For interpolated points, a splitting fraction $w_i(\tau)$ (analogous to $\eta_Z(\tau)$) determines probability mass allocation.

Key algorithmic features:

• At each round (quantile level), a piecewise-linear programming problem is solved:

$\min_{b} \sum_{i}(X_i - Z_i^\top b)^+,$

subject to constraints derived from data stratification by the hyperplane.

• The solution is updated at breakpoints, typically via line search as new observations interpolate with the current estimate.
• The algorithm is numerically robust and avoids grid dependence or ambiguous pivoting, reliably converging in moderate- to high-dimensional settings.

3. Theoretical Foundations: Consistency and Weak Convergence

Under regularity conditions—bounded covariates, continuous and bounded survival/censoring distributions, Lipschitz continuity of $\beta_0(\tau)$—the integral equation estimator enjoys strong large-sample properties:

• Uniform consistency over any interval $[l, u]$ within the identifiability region:

$$\sup_{\tau\in[l,u]} \|\hat\beta(\tau)-\beta_0(\tau)\| \to 0 \quad \text{a.s.}$$

• Weak convergence to a Gaussian process:

$$n^{1/2}\left\{ \hat\beta(\tau)-\beta_0(\tau) \right\} \rightsquigarrow \mathbb{G}(\tau)$$

for $\tau$ in $[l,u]$. Technical arguments use empirical process theory and the Grönwall inequality for functional control.

4. Online Inference: Multiplier Bootstrap and Sequential Inference

To deliver valid statistical inference—including confidence bands for the entire coefficient process—the method employs a multiplier resampling (bootstrap) approach:

• The estimating equation is perturbed using independent unit-mean, unit-variance random weights (e.g., i.i.d. exponentials), and the perturbed estimator is recomputed.
• Theorem 3 guarantees that, conditionally on the data, the distribution of the bootstrap estimator matches the asymptotic law of the original estimator. This enables:
  • Construction of uniform confidence bands for $\beta_0(\tau)$ over $\tau \in [l,u]$.
  • Compatibility with online or real-time updating of inferential quantities as new data are observed.

5. Empirical Performance: Simulation and Clinical Application

Simulation studies mimic clinical settings with moderate censoring (~32%) and nonconstant covariate effects:

• The estimator achieves negligible bias and better performance at high quantiles than alternatives such as Portnoy's pivoting method.
• Efficiency rivals that of Peng and Huang's grid-based approach, but with increased numerical stability.
• The PLMIN implementation is fast and avoids failures present in other methods as the number of covariates or censoring increases.

A clinical application to the Mayo primary biliary cirrhosis dataset (n=416, 61.5% censoring) demonstrates:

• Ability to resolve both constant and time-varying covariate effects (e.g., varying effect of log(prothrombin time)).
• Computed trimmed mean (integrated quantile coefficient) effects for effect summary.
• Enhanced interpretability and flexibility compared to accelerated failure time and proportional hazards models.

6. Comparative Perspective and Methodological Advantages

The estimator generalizes classical procedures:

• Reduces exactly to the Koenker–Bassett quantile estimator when there is no censoring.
• Reduces to the Kaplan–Meier estimator in the $k$-sample case (no covariates) via cadlag inversion of the empirical survival function.

Key advantages over other approaches:

• No need for probability grid discretization or dependence on intricate pivoting.
• No additional modeling or estimation for censoring-time distribution required.
• Robustness to covariate-dependent censoring, accommodating continuously-distributed covariates.
• Numerically reliable and efficient for fully online estimation of the quantile regression process.

Limitations include potential issues with solution uniqueness and nonregularity around zero-density regions of the survival distribution, where classical asymptotic behavior may fail. Outside such regions, the estimator remains reliable.

7. Extensions and Related Work

The theoretical framework and computational algorithm also have connections to:

• Extreme regression quantile methods [Portnoy & Jurečkov (1999); Chernozhukov (2005)].
• Extensions allowing discontinuities and zero-density intervals, with careful reinterpretation of estimation targets.
• Possible integration into more general online and real-time analytic frameworks due to the estimator's functional, recursive structure.

In summary, the online inference method derived from quantile calculus and PLMIN provides a rigorous, efficient, and fully functional approach to censored quantile regression, unifying and extending the capabilities of prior methods and offering strong theoretical guarantees and practical reliability for survival data analysis (1010.0514).