I'm wondering if this density has a sufficient statistic to summarize an independent and identically distributed sample of random variables $X_1,X_2,...,X_n$ of this density.
The density is:
After some computation, arrived to:
But with this expression, it seems impossible to resume the information in a statistic because of the exponent of the middle factor. There, the $x_i$'s will allways be separated, but together (each one) with the parameter ($\theta$). So it will be impossible to factorize this expression (to use the Halmos Factorization Theorem), meaning that there is no sufficient statistic. Am I concluding correctly, or i'm missing something?