[Paper Review] Quantile regression in transformation models

Dorota M. Dabrowska|ArXiv.org|Nov 21, 2005

Statistical Methods and Inference18 references18 citations

TL;DR

This paper develops quantile regression methods for semiparametric transformation models with censored data, proposing estimators for conditional quantiles and constructing confidence sets via weak convergence of empirical processes. The key contribution is the asymptotic normality of the quantile process under censoring, enabling valid inference in survival analysis with unknown transformation functions.

ABSTRACT

Conditional quantiles provide a natural tool for reporting results from regression analyses based on semiparametric transformation models. We consider their estimation and construction of confidence sets in the presence of censoring.

Motivation & Objective

• To extend quantile regression to semiparametric transformation models under right-censoring, enabling inference on conditional quantiles.
• To develop estimators for conditional quantiles that account for unknown transformation functions and censoring mechanisms.
• To construct valid confidence sets for quantile functions using weak convergence of empirical processes.
• To establish the asymptotic distribution of the quantile process under censoring, ensuring inferential validity.

Proposed method

• Models the conditional distribution of failure time T given covariates Z using a transformation model F(t|z) = F(Γ(t), θ|z), where Γ is an unknown increasing transformation.
• Expresses conditional quantiles as Q(p|z) = Γ⁻¹(exp(−θᵀz)G⁻¹(p)), linking quantiles to the regression coefficient θ and transformation Γ.
• Uses a time-transformed martingale approach to derive the asymptotic distribution of the quantile process, accounting for censoring via weighted empirical processes.
• Applies weak convergence theory to show that the scaled difference between estimated and true quantile processes converges to a Gaussian process.
• Derives confidence sets for the quantile function using the limiting Gaussian process and empirical process approximations.
• Employs integration by parts and Gronwall's inequality to control estimation error in the presence of estimated transformation functions.

Experimental results

Research questions

• RQ1 How can conditional quantiles be estimated in transformation models when data are subject to right-censoring?
• RQ2 What is the asymptotic distribution of the quantile process in censored transformation models?
• RQ3 How can confidence sets for the conditional quantile function be constructed under censoring?
• RQ4 What is the impact of estimating the unknown transformation function on the asymptotic behavior of the quantile process?

Key findings

• The quantile process, properly scaled, converges weakly to a mean-zero Gaussian process under the null model, enabling asymptotic inference.
• The limiting distribution of the quantile process is derived using a time-transformed martingale representation, accounting for censoring.
• The difference between the estimated and true quantile processes converges to zero in probability, ensuring consistency.
• Confidence sets for the conditional quantile function can be constructed using the limiting Gaussian process and empirical process approximations.
• The convergence of the quantile process is robust to estimation of the transformation function, as shown via Gronwall's inequality and uniform error bounds.