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?
Best Answer
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 $\alpha$.