# minimal sufficient statistics of 1-parameter Gamma distribution

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.

• So-called "curved" exponential families usually require as many dimensions in the sufficient statistics as the natural ones. Apr 13, 2021 at 19:40
• 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. Apr 14, 2021 at 1:34