Summary findings from the simulation study
From the simulations, the naive estimator has substantial bias in some scenarios. The approximate UMVCUE has the smallest biases, being practically unbiased. The multiple iterations bias subtracted estimator is slightly more biased than the UMVCUE, mostly overcorrecting for bias. Apart from that, the two estimators have similar properties. They are approximately unbiased but in most scenarios, they have higher RMSEs than the naive estimator. This is because, from the boxplots, their estimates are approximately symmetric around the true value whereas when the naive estimator is biased, more than half of its estimates are below the true value but its smallest log hazard ratio is approximately equal to smallest log hazard ratios for the UMVCUE and the multiple iterations bias subtracted estimator. Therefore, in clinical trials where unbiased estimators are required, we consider the UMVCUE and multiple iterations bias subtracted estimator to be better than the naive estimator, with the former slightly better. The single iteration bias subtracted estimator does not eradicate bias fully, retaining substantial bias if the selected treatment is not the most effective, but it has smaller RMSEs compared to the UMVCUE and multiple iterations bias subtracted estimator and so it may be preferable in clinical trials where some overestimation is acceptable. The shrinkage estimator does not eradicate much bias but it has smallest RMSEs in most scenarios.Findings for the case of two experimental treatments in stage 1 are similar (Supplementary Material, Tables Simulations results when estimation is based on the logrank score statistic are summarized in the Supplementary Material (Tables 