If $\{E(\lambda)\}_{\lambda > 0}$ denotes the exponential family, then $E(\frac{1}{\theta}), \theta > 0$ is an alternative parameterization. I think a sufficient statistic for $\lambda$ is also sufficient for $\theta = \frac{1}{\lambda}$.
Do I prove this using contradiction? My thinking is if the statistic were not sufficient for $\theta$, then it wouldn't have been sufficient for $\lambda$ in the first place. If that's right, I'd like to answer my own question.