I’m revising for an exam in Mathematical Statistics and I have found the following problem in one of the previous exams (I don’t know the source):
Let $X_1,…,X_n$ be a random sample that follows the Weibull distribution with parameters $\delta=\theta^{-1/6}$, $\eta=6$, where $\theta$ is unknown. Prove that $T=\sum_{i=1}^n X_i^6$ is sufficient for $\theta$.
Now, according to my book and notes on Neyman’s factorization criterion, I need to prove that $f(x;\theta)=G(t,\theta)H(x)$. So I went ahead and calculated:
$$f(x;\theta)=\prod_{i=1}^n \frac{\eta}{\delta}\cdot\bigg(\frac{x_i}{\delta}\bigg)^{(\eta-1)}\cdot e^{-(x_i/\delta)^\eta}$$
which, if I’m correct is:
$$f(x;\theta)=6\cdot \theta\cdot \prod_{i=1}^n x_i^5 \cdot e^{-\theta x_i^6}$$
and this is where I get stuck. I can make some changes and produce:
$$f(x;\theta)=6\cdot\theta\cdot \prod_{i=1}^n e^{-\theta x_i^6} \cdot e^{\ln x_i^5} = 6\cdot \theta \cdot e^{-\theta \sum_1^n x_i^6}\cdot e^{\sum_1^n \ln x_i^5}$$
So, I have the $\sum_1^n x_i^6$ but I’m still left with the last part that I don’t understand. Can someone explain to me what I’m doing wrong??