Finding minimal sufficient for the following distribution???

Question

Hi I was working on finding the minimal sufficient for the following distribution $$f(x|\theta)=\theta^{-1}x^{\frac{1-\theta}{\theta}}I(0\le x\le 1),\,\,\,\,\,\,\theta>0$$ By factorization theorem and by checking the ratio as constant function about $\theta$ I found $\prod_{i=1}^n x_i$ as a minimal sufficient statistic. Could anyone help me whether I am right or wrong? If anyone want to have a look at my calculation, I can post it here. Thanks in advance

Answer 1

You are right, by using the factorization criteria you got $$ \mathcal{L}(\theta ; X) = \prod_{i=1}^n \frac{1}{\theta} x_i ^{\frac{1-\theta}{\theta}}I(0 \le x_i \le 1 ) = \frac{1}{\theta^n} \left(\prod_{i=1}^n x_i\right) ^{\frac{1-\theta}{\theta}}\prod_{i=1}^nI(0 \le x_i \le 1 )\, , $$ so $g(\theta; X) = (\prod_{i=1}^n x_i) ^{\frac{1-\theta}{\theta}} $, hence $h(X) = \prod_{i=1}^n x_i$ is the required MSS.