Let $X_1, X_2 , \dots, X_n$ be i.i.d observations from $N(\theta, \theta)$ (where $\theta$ is the variance). I intend to find the minimal sufficient statistics (M.S.S) for the joint distribution.
My steps ultimately lead me to the following:
$\dfrac{f_\theta(x)}{f_\theta(y)} = \text{exp}{ \left\{ \dfrac{-1}{2\theta} \left[ \left(\sum x_i^2 - \sum y_i^2 \right) + 2\theta \left( \sum y_i - \sum x_i \right) \right] \right) }$.
My M.S.S are therefore $\left( \sum x_i^2 , \sum x_i \right)$
Am I on track?