Suppose $X_1,X_2,\ldots,X_n$ be a random sample of distribution with probability density function
$$f(x\mid\theta) = \theta x^{\theta-1},\quad 0\lt x \lt 1,\quad 0\lt \theta \lt \infty$$
how can i find Sufficient statistics of parameter $θ$?
Need some guidance on this. Not very sure how to start...
The joint density is $$ \prod_{i=1}^n (\theta x_i^{\theta-1}) = \theta^n\left(\prod_{i=1}^n x_i\right)^{\theta-1}. $$ Since this depends on $(x_1,\ldots,x_n)$ only through their product, the product is sufficient.
If you can put the joint density into a form in which it's a product of two things, and one of those factors depends on $(x_1,\ldots,x_n)$ only through a particular statistic $T$, and the other factor does not depend on $\theta$, then $T$ is sufficient. That is Fisher's factorization criterion. That's probably quicker in this case than going back to the definition of sufficiency (which in this case would say that the conditional distribution of $(X_1,\ldots,X_n)$ given $X_1\cdots X_n$ does not depend on $\theta$ (Notice: here I'm using capital $X$; above I use lower-case $X$; I mention this because it's not unusual to see people who post in forums like this neglecting the distinction).