Simulation of remission and after remission times under specific conditions

This is basically a data generation problem.

Say, t is an exponential lifetime with mean two years. tr is the remission time and ts is the after remission time. So, t=tr+ts. I need to simulate these quantities for 100 patients. Now in real life, usually when tr is smaller, that is when the patients remit quickly, their after remission time ts are longer (so that the lifetimes t are also longer). When tr are longer, that is the patients remit slowly, their after remission times ts are shorter.

One thing is known that the distribution of the lifetimes, t is exponential with mean two years. I don't have any information about their exact relationship but I know, this is the scenario in real life. I need to simulate data under this scenario because I want to see the performance of an estimator under these conditions. So any feasible assumption regarding how they are correlated is absolutely okay for me. But I cannot understand how to generate data from the above mentioned scenario. If some sort of randomness can be put along with maintaining the conditions, that will be great!

Your suggestion will be a great help.


TS Contributor
It seems that you always need to have more assumption on the distributions of either \( t_s \) or \( t_r \).

One very naive way I could think of is that first you simulate the exponential distributed \( t \).

Next you need to assign a distribution with support \( (0, t) \), e.g. \( t\text{Uniform}(0, 1) \) or \( t\text{Beta}(a, b) \) for let say \( t_r \). Subsequently you can obtain \( t_s = t - t_r \).