What is the sufficient statistic for a beta distribution?

Let {$$X_1,\ldots,X_n$$} be a random sample from the $$beta(\alpha,\beta)$$ distribution.

Below is the beta distribution with the parameters referred to:

$$f_X(x;\alpha,\beta)=\frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)(\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}$$

How do I find a sufficient statistic for (a) $$\alpha$$ when $$\beta$$ is known (b) $$\beta$$ when $$\alpha$$ is known?

My approach:

The factorization theorem is $$\prod_{i=1}^n f(x_i ; \alpha,\beta)= \frac{1}{B ^n( \alpha, \beta)} \left( \prod_{i=1}^n x_i \right) ^ {\alpha-1} \left(\prod_{i=1}^n(1-x_i)\right)^{\beta-1}$$ thus, considering $$T(x) =\prod_{i=1}^n x_i^{-1} ( 1 - x_i)^{-1}$$ one sees that $$g_{\theta}(T(x)) = \frac{1}{B ^ n( \alpha, \beta)} \left( \prod_{i=1}^n x_i ( 1 - x_i) \right) ^ {\alpha\beta}$$

I am not sure if I am on the right track. How do I proceed from here in finding a sufficient statistic for $$\alpha$$ when $$\beta$$ is known and $$\beta$$ when $$\alpha$$ is known?

• Recall how the function $g$ is defined in the Factorisation theorem. – StubbornAtom Jan 17 at 10:22

When $$\beta$$ is known, you can take $$T(x) = \prod _i x_i$$ and $$g_\alpha(T(x)) = \frac{1}{B^n(\alpha,\beta)} \left(\prod_i x_i \right)^{\alpha - 1}$$ in the factorization theorem for $$\alpha$$.