complete sufficient statistic exercise

Question

I have to find complete sufficient statistic of the following pdf

$$f(x|\theta)=\frac{\theta}{(1+x)^{(1+\theta)}},\quad 0<x<\infty,\theta>0.$$

My Attempt:

The joint density

$$f(\mathbf x|\theta)=\prod_{i=1}^{n}\frac{\theta}{(1+x_i)^{(1+\theta)}}$$

$$=\theta^n\prod_{i=1}^{n}\exp[-(1+\theta)\log(1+x_i)]$$ $$=\theta^n\exp[-(1+\theta)\sum_{i=1}^{n}\log(1+x_i)]$$

So, $\sum_{i=1}^{n}\log(1+x_i)$ is complete sufficient.

But it seems to me I am wrong. What will be the correct answer ?

Answer 1

Your derivation is correct. You have a regular exponential family, so the factorization theorem gives $\sum_i \log (1+x_i)$ is sufficient, and complete since it is a regular exponential family.