Suppose $X_1 \sim Pois(\lambda_1), X_2 \sim Pois(\lambda_2), X_3 \sim Pois(\lambda_1+\lambda_2)$. Separately I can find a sufficient, complete and minimal statistic for each of them. But considering the joint distribution - is it still possible?
Does it depend on which parameters I consider? I do not see how we can find a sufficient statistic if we consider only $(\lambda_1, \lambda_2)$ (can I ignore the information given by $X_3$?), but is it possible to reparametrize and consider, say $(\lambda'_1 = \lambda_1, \lambda_2' = \lambda_2, \lambda_3' = \lambda_1+\lambda_2)$.
In general what is the interaction between reparameterization and sufficiency?