Gamma Distribution Sufficient Statistics

Question

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!

Answer 1

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