Estimating Baseline Survival Function in the Proportional Hazards Model Under Monotone Hazards

In biomedical studies where patient risk is known to increase over time, estimating the baseline survival function under the Cox proportional hazards (PH) model becomes particularly challenging when data involve delayed entry (left truncation) and incomplete follow-up (right censoring). This paper proposes a Breslow-type estimator for left-truncated and right-censored data under a monotone hazard assumption. By incorporating the Cox regression coefficients into Tsai's monotone maximum likelihood estimator (MLE), we develop a covariate-adjusted monotone MLE that generalizes both Tsai's univariate approach and Lopuhaä's monotone estimator, which is restricted to right-censored data. Theoretically, we establish strong consistency of the estimator and derive its asymptotic distribution at a fixed point. Empirically, simulations confirm that our approach remains numerically stable with sparse early-time data, yields nearly unbiased survival estimates even in small samples, and achieves consistent efficiency gains from baseline covariate adjustment. We illustrate the method with an application to the Channing House data.