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Suppose you are using the factorization theorem to find a sufficient statistic. Let us say that we have a negative sign in front of $T(x)$. How do you know whether or not to "absorb" a negative sign into a sufficient statistic or any other sign? So instead of $T(x)$ we could have $$\widetilde T(x):=−T(x) \>.$$

For example, suppose you have $$g(T(x)| \theta) = e^{-T(x)}$$ in the factorization theorem. How do you know whether or not to include the negative sign in $T(x)$?

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They are both sufficient and we can say the same thing in a more general context: If $T$ is a sufficient statistic for some parameter $\theta$ than any almost-surely invertible transformation $U = h(T)$ is also sufficient for $\theta$. – cardinal Apr 8 '12 at 20:39
@cardinal: So if $\bar{X}$ is sufficient for $\mu$ then how can $-\bar{X}$ be sufficient for $\mu$? If $\mu$ is known to be positive then how can $-\bar{X}$ be sufficient for $\mu$? – ross Apr 9 '12 at 4:18
Hi ross, you should be able to click the "Add comment" link to respond to the comments you see. It might help to think about two things: (1) The statement of the Factorization Theorem itself and (2) The definition of sufficiency vis a vis conditioning on a sufficient statistic. I can post an answer with some more details, if that would help. – cardinal Apr 9 '12 at 14:24

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