# Gamma Distribution Sufficient Statistics

I've been asked to show the gamma distribution can be written in the form $$p(x|\alpha, \beta) = f(x) g(\alpha, \beta) e^{h(\alpha,\beta)^T T(x)}$$ where $$T(x)$$ is a sufficient statistic.

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I have done the following:

$$p(x|\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x} = \frac{\beta^\alpha}{\Gamma(\alpha)} exp[(\alpha-1)\log(x) - \beta x]$$ and so identified $$g(\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}, f(x)=1, h(\alpha,\beta) = \begin{pmatrix} \alpha-1 \\ \beta \end{pmatrix}, T(x) = \begin{pmatrix} \log{x} \\ x \end{pmatrix}$$ which I believe is correct.

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However, I do not understand why I cannot take the original form and identify $$g(\alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}, f(x) = x^{\alpha-1}, h(\alpha,\beta) = - \beta, T(x) = x$$.

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My question is essentially in two parts:

1, Presumably this is because the question demands that $$T(x)$$ be a sufficient statistic. Now I understand that a sufficient statistic is basically all the information required so that we can compute the parameters $$\alpha$$ and $$\beta$$. Is that correct? Is there a better way to think about it?

2, Assuming that is correct, my problem reduces to determining $$T(x)$$ and justifying that my first approach is correct. I have read online that $$T(x)$$ is determined by finding the conditions necessary for $$\frac{p(x|\alpha,\beta)}{p(y|\alpha,\beta)}$$ to be independent of both $$\alpha$$ and $$\beta$$. I don;t really understand why though? Can someone explain?

Moving on, we get

$$\frac{p(x|\alpha,\beta)}{p(y|\alpha,\beta)} = \frac{\frac{\beta^\alpha}{\Gamma(\alpha} x^{\alpha-1}e^{-\beta x}}{\frac{\beta^\alpha}{\Gamma(\alpha)} y^{\alpha-1}e^{-\beta y}} = \frac{x^{\alpha-1}}{y^{\alpha-1}} e^{-\beta(x-y)}$$

At this point is seems to me that if I choose $$x=y$$ I can make this independent of both $$\alpha$$ and $$\beta$$ and so the sufficient statistic should just be $$T(x)=x$$ but this is definitely wrong.

I am just learning this by myself and haven't found many good resources online. If someone could fix my problem and also direct me to some good online notes or a good book I'd really appreciate it.

Thanks!

I do not understand why I cannot take the original form and identify $$g(\alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}, f(x) = x^{\alpha-1}, h(\alpha,\beta) = - \beta, T(x) = x$$
The reason for this decomposition to be incorrect is that $$x^{\alpha-1}$$ depends on the parameter $$(\alpha,\beta)$$, thus that there is no separation from the parameter.
That $$T(\cdot)$$ is a sufficient statistic proceeds from the factorisation theorem: $$p(x|\alpha, \beta) = f(x) g(\alpha, \beta) e^{h(\alpha,\beta)^\text{T} T(x)}$$shows that the density of $$X$$ factorises as a function of $$x$$, $$f(x)$$, and a function of $$T(x)$$ only, namely $$g(\alpha, \beta) \exp\{h(\alpha,\beta)^\text{T} T(x)\}$$
I have read online that $$𝑇(𝑥)$$ is determined by finding the conditions necessary for $$𝑝(𝑥|𝛼,𝛽)/𝑝(𝑦|𝛼,𝛽)$$ to be independent of both $$𝛼$$ and $$𝛽$$
This is incorrect: the ratio is independent of the parameter value when $$T(x)=T(y)$$, again by the factorisation theorem
• Hi Xi'an, thanks for your reply. I've added the tag. I can see that $x$ and $\alpha$ are connected from the $x^{\alpha-1}$ term and thus my $f(x)$ would actually be $f(x,\alpha)$ which is not allowed. However, I do not understand the rest of your post. This is probably a more serious misunderstanding of how to calculate sufficient statistics. Could you show me the details of how to find $T(x)$ or do you have a link I could read up on them that explains them clearly? Thanks. – user11128 Feb 13 '19 at 13:04