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?

  • $\begingroup$ Sufficiency does not depend on the selected (re)parameterisation of the model, cf the generic definition of exponential families and the subsequent definition of natural parameters. (Sufficiency only makes sense within exponential families.) $\endgroup$
    – Xi'an
    Oct 19 '20 at 9:24

You have three random variables, suppose independent (you did not say, but I assume that was the intention. Write down the likelihood function, a product of three Poisson pmf's. Then use the factorization theorem. You should be able to conclude that there is no sufficient reduction , that is $T=(X_1, X_2, X_3)$ is sufficient, and no sufficient reduction can be found (you can check that $T$ is minimal sufficient).

But the sufficient statistic $T$ is not complete, that is, $\DeclareMathOperator{\E}{\mathbb{E}} \E(X_1+X_2-X_3)=0$. In fact, the distribution of $(X_1, X_2, X_3)$ forms a curved exponential family.

As for the question in the title: Results about sufficiency, minimal sufficiency, completeness, do not depend on the parametrization used.


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