Suppose, X,Y,Z are three positive independent variable. The cumulative distribution of X is as follows:
\begin{equation} F_X(t) = \left\{ \begin{array}{cc} F_1(t) & \mathrm{if\ } 0<t<\tau_1 \\ F_2(t) & \mathrm{if\ } \tau_1<t<\tau_2 \\ F_3(t) & \mathrm{if\ } \tau_2<t<\infty \\ \end{array} \right. \end{equation}
Siilarly Y and Z has distributions like this in the same intervals but are not identical to X.
Also, cumulative distribution function:
$$ F_1(t), F_2(t) \ and\ F_3(t) $$ are such function so that $$F_1(\tau_1)=F_2(\tau_1) $$ and $$F_2(\tau_2)=F_3(\tau_2)$$ T is the minimum of X, Y, and Z. My question is: How to simulate T from X,Y and Z so that we can get the value of T as well as an indication of where it has come from? indication = 1 if the minimum has come from X, indication = 2 if minimum has come from Y, and indication = 3 if the minimum has come from Z. Sample size should be n, and it is not specific how many samples there should be between $\tau_{i-1} $ and $ \tau_{i}$.
I was trying to simulate 3 uniform distributions by cutting them $$ F(\tau_1), F(\tau_2) $$ but problem is that I am not getting the same number of samples in each boundary of the tau. If the sample size is 1, then it's easy, but when it's not 1 and not specific for different intervals, it's stuck.
For sake of simplicity i was doing for exponential distribution with different parameters.