# Gamma distribution: ratio of 2 CSS not containing $\beta$

Let $$X_1,...,X_n$$ be iid and follow $$Gamma(\alpha, \beta)$$, where $$f(x,\alpha, \beta)=\frac{x^{\alpha-1}e^{-x/\beta}}{\Gamma(\alpha)\beta^\alpha}$$ I already showed that $$\overline{X}$$ and $$X^*=\left[ \prod_{i=1}^n X_i \right]^\frac{1}{n}$$ are complete and sufficient statistics for $$(\alpha, \beta)$$. I am to show that the distribution of the statistic $$T=\frac{\overline{X}}{X^*}$$ does not depend on $$\beta$$ but am unsure how to do so. Any help would be greatly appreciated.

• What do you mean by "does not contain $\beta$"? When I look at the formulae, I see no $\beta$... Commented Apr 12, 2020 at 2:16
• I edited the wording of the problem a little bit; hopefully that clears it up. The $X_i$'s follow $Gamma(\alpha, \beta)$ and I need to show that the statistic $T$ is ancillary for $\beta$. Commented Apr 12, 2020 at 2:25
• Please mention the parameterization you are using, i.e. the pdf of Gamma(a,b). Commented Apr 12, 2020 at 7:29
• As a hint: try a change of variable from $x$ to $y=x/\beta$. If you solve for the distribution of $y$, you will see that it doesn't have $\beta$ as a parameter. Then show that $\bar{Y} / Y^* = \bar{X}/X^*$, and you're essentially done. Commented Apr 12, 2020 at 14:35
• @jbowman that was a giant help. Thank you so much! Commented Apr 12, 2020 at 15:54

## 1 Answer

Let $$Y=X/\beta$$. Then, $$f_Y(y)=f_X(y\beta)\beta=\frac{(y\beta)^{\alpha-1}e^{-(\beta y/\beta)}}{\Gamma(\alpha)\beta^\alpha}$$ $$=\frac{y^{\alpha-1}e^{-y}}{\Gamma(\alpha)}$$ Note that the distribution of Y does not depend on $$\beta$$. Now consider $$\overline{Y}=\frac{\sum Y_i}{n}=\frac{\sum X_i/\beta}{n}=\frac{1}{\beta}\frac{\sum X_i}{n}$$ $$Y*=\left[\prod_{i=1}^n Y_i\right]^{1/n}=\left[\prod_{i=1}^n X_i/\beta \right]^{1/n}=\frac{1}{\beta}\left[\prod_{i=1}^n X_i\right]^{1/n}$$ Then, when taking the ratio $$\frac{\overline{Y}}{Y*}$$, note that the $$\frac{1}{\beta}$$ pieces of both statistics cancel, and you're left with $$\frac{\overline{X}}{X*}$$. This tells us that the statistic $$T=\frac{\overline{X}}{X*}$$ does not depend on $$\beta$$.