Asymptotic properties of the MLE in distributional regression under random censoring
Gitte Kremling, Gerhard Dikta
TL;DR
This work analyzes maximum likelihood estimation for distributional regression under random right censoring with covariates. It derives a practical partial likelihood that isolates the censored contributions and establishes identifiability via extended Kullback–Leibler information, proving almost-sure consistency $\hat{\vartheta}_n\to\vartheta_0$ and asymptotic normality $\sqrt{n}(\hat{\vartheta}_n-\vartheta_0)\Rightarrow Z\sim N(0,\Sigma^{-1})$ under standard regularity conditions. The covariance matrix $\Sigma$ incorporates censoring via two components, and an asymptotically linear representation is provided. A simulation study demonstrates favorable finite-sample performance, showing decreasing MSE with sample size and robustness to substantial censoring. The results extend censored distributional regression methods to random covariates and offer rigorous inference guarantees for applied survival-type analyses.
Abstract
The aim of distributional regression is to find the best candidate in a given parametric family of conditional distributions to model a given dataset. As each candidate in the distribution family can be identified by the corresponding distribution parameters, a common approach for this task is using the maximum likelihood estimator (MLE) for the parameters. In this paper, we establish theoretical results for this estimator in case the response variable is subject to random right censoring. In particular, we provide proofs of almost sure consistency and asymptotic normality of the MLE under censoring. Further, the finite-sample behavior is exemplarily demonstrated in a simulation study.