Please could you help check my approach to finding a one dimensional sufficient statistic.
Here is the problem.
$X_1,...,X_n$ is a random sample with each $X_i$ having the pdf $f(x;\theta)=2\theta^2x^{-3} $ and $0<\theta\leq x<\infty$
Approach
$L(\theta;x)= \prod_{i=1}^n 2\theta^2x^{-3}_i $
$L(\theta;x)= 2^n \theta^{2n}x^{-3n}_i $
$L(\theta;x)= [2^n] [\theta^{2n}][x^{-3n}_i] $
So by factorisation theorem, $x^{-3n}_i$ is a sufficient statistic.
Is this right?