I have some survival times which are exponentially distributed for two groups G1 (treatment) and G2 (control). The data are censored with a censoring distribution given by h(c), so I only observe:
1) times to event X - defined as Xi = min(Ti, Ci), where Ti is the true survival time and Ci is the censoring time
2) censoring status Z - where Zi = 0 if Ti was censored and Zi = 1 if Ti was not censored
The survival times and censoring times are independent and drawn iid from their respective distributions.
I'm trying to conduct a hypothesis test in which I want to determine whether the median survival under treatment is better than control.
I have calculated a maximum likelihood estimate for the rate parameter of the exponential distribution of Ti under the joint distribution of Xi and Zi. The expression is identical for both treatment and control groups. Asymptotically, the distribution of the MLE should converge to Normal(true value, s^2). The variance of the distribution of the MLE (s^2) can be estimated by bootstrapping.
Given this knowledge, what test statistic can I calculate that will allow me to test my hypothesis and what test do I perform? I think this should be a two-sample, one-tailed z-test but I'm not sure.
Thanks!