A pair of sufficient statistics for a gamma

Question

You have independent positive observations $X_1, \dots, X_n$ from a Gamma distribution with density proportional to $x^{\alpha - 1} e^{-\beta x}$ , where both the shape $\alpha$ and the rate $\beta$ are unknown.

Find a pair of statistics that are jointly sufficient for $(\alpha, \beta)$ .

Accepted Answer

The plan. Factor the joint density into a parameter-piece times a data-only piece, and see which data summaries the parameter-piece needs. Multiply the densities. Up to a constant that depends only on \alpha, \beta (the normalizing factor), f(x; \alpha, \beta) \propto \prod{i=1}^n xi^{\alpha - 1} e^{-\beta xi} = \Big(\prodi xi\Big)^{\alpha - 1} \exp\!\Big(-\beta \sumi xi\Big). Turn the product into a sum of logs, since \big(\prodi xi\big)^{\alpha-1} = \exp\!\big((\alpha - 1)\sumi \log xi\big): f(x; \alpha, \beta) \propto \exp\!\Big((\alpha - 1)\sumi \log xi - \beta \sumi xi\Big). The paramete…

A pair of sufficient statistics for a gamma

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