If $x_i \sim Gamma(\alpha, \alpha)$, are the minimal sufficient statistics still $\Pi_i x_i$ and $\sum_i x_i$ (same as when $x_i \sim Gamma(\alpha, \theta)$ where $\alpha \neq \theta$)? My reasoning is as follows:
For $x = (x_1, \cdots, x_2)$ and $y = (y_1, \cdots, y_2)$ each being a sample from $Gamma(\alpha, \alpha)$, $\displaystyle \frac{f(x|\alpha)}{f(y|\alpha)} = \frac{(\Pi_i x_i)^{\alpha-1}}{(\Pi_i y_i)^{\alpha-1}} \exp(-\frac{\sum_i x_i - \sum_i y_i}{\alpha})$
To make it independent of $\alpha$, seems it requires (1) $\Pi_i x_i = \Pi_i y_i$ (2)$\sum_i x_i = \sum_i y_i$, so the minimal sufficient statistics are $\Pi_i x_i$ and $\sum_i x_i$
However it's a bit surprising to me that we still need two statistics in this reduced parameter case.