Hey DavidEriksson.

If you wish to use a pivotal quantity, then I would be looking at using the chi-square pivotal quantity for estimating whether the parameters are the same or not.

Recall that the pivotal quantity doesn't depend on the parameters or its distribution and what you are doing is the opposite where you are deriving specific sampling distributions to test your hypotheses: you can take this approach if you wish but its not the same as using pivotal quantities like the Normal Distribution or the chi-square distribution.

The reason also for using chi-square is that the exponential parameters must be greater than zero, so using a Normal pivotal quantity wouldn't work.