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I am reading Robert V. Hogg Introduction to Mathematical Statistics 6th Version page 409, second paragraph.

$X_1, X_2$ is a random sample from a Gamma $\text{G}(\alpha,\beta)$ distribution with known parameter $\alpha>0$ and unknown parameter $\beta$. The ratio $$Z=\dfrac{X_1}{X_1+X_2}$$ has a Beta $\text{B}(\alpha,\alpha)$ distribution that is free of $\theta$. Therefore $Z$ is a ancillary statistic.

My question is: why $Z$ is free of $\beta$?

In the book p155, the authors showed that the pdf of $Z$ is $$f(z)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}z^{\alpha-1}(1-z)^{\beta-1},\quad 0<z<1$$ Or you can ref to this one. This is the pdf for the Beta $\text{B}(\alpha,\beta)$ distribution but the $\beta$ or $\theta $ is still there, why $Z$ is free of $\beta$ then?.

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There is either a typo in giving the pdf of $Z$ or you are confusing with the general definition of a Beta $\text{B}(\alpha,\beta)$ as it should be a Beta $\text{B}(\alpha,\alpha)$ distribution. For instance, your link shows why the ratio of two Gamma $\text{G}(\alpha_i,\beta)$ variates is a Beta $\text{B}(\alpha_1,\alpha_2)$ variate. (This reference can be confusing as it uses $\alpha$ and $\beta$ in the opposite of the standard way: $\alpha$ is the scale there!)

The reason why $Z$ does not depend on $\beta$ is that, when $$X_1,X_2\sim\text{G}(\alpha,\beta)$$ then $$Y_1=\beta X_1,Y_2=\beta X_2\sim\text{G}(\alpha,1)$$ since $\beta$ is a scale factor. Therefore $$Z=\dfrac{X_1}{X_1+X_2}=\dfrac{\beta X_1}{\beta X_1+\beta X_2}=\dfrac{Y_1}{Y_1+Y_2}$$ does not depend on $\beta$.

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