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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.

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    $\begingroup$ So-called "curved" exponential families usually require as many dimensions in the sufficient statistics as the natural ones. $\endgroup$
    – Xi'an
    Apr 13, 2021 at 19:40
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    $\begingroup$ Please add the self-study tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. $\endgroup$ Apr 14, 2021 at 1:34

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