Confidence interval of 2 exponential random variables with different paramerts

Question

So i have 2 independent samples, $X_1,X_2...,X_n$, where $X_i$~exp($\lambda$), and $Y_1,Y_2...,Y_n$, where $Y_i$~exp($2\lambda$).
I want to find a confidence interval for $\lambda$ with confidence level of $1-\alpha$.

So far using MLE I've found an estimator for $\lambda$, $\hat{\lambda}=\frac{n\overline{X}+2m\overline{Y}}{n+m}$, but im stuck finding the pivotal quantity.
I would be happy to receive ideas or references about how it can be done.

Answer 1

Community wiki answer so the question can be marked as answered:

As pointed out by Henry in a comment, you can combine the samples into one by transforming $Y$ to $Z=2Y\sim\exp(\lambda)$.