The following probability density function:
$$f(x)= \frac {(\log\alpha)\alpha^x} {\alpha-1} ;\qquad 0<x<1,\quad \alpha>1$$
Using factorization theorem the sufficient statistic would be $T=\sum_{i=1}^n x_i$
If the number of sufficient statistics = number of parameters. Thus the completeness is exists. In our case the completeness is exist. I want to proof the following.
$$E(g(T))=0 \quad \text{ then }\quad g(T)=0 $$
Thus $T$ is complete sufficient statistic.
First we want to find the distribution of sufficient statistic. How can we find the distribution of sufficient statistic $T=\sum_{i=1}^n x_i$? Also, Can we use Laplace transformation for variable its domain [0, 1] Not [0, infinity[.
Visit for more details about Laplace transformation. Any assistance would be greatly appreciated.