Let we have $n$ independent and identical random variables from gamma distribution with parameters $2$ and $\theta$, ie $G(2,\theta)$ where $n$ is greater than or equal to $2$. Check whether the following are sufficient and complete:
- $\overline{X}$
- $\overline{X}^2+3$
- $(X_1,\sum_{i=2}^{n} X_i)$
- $(X_1, \overline{X})$
I think all of them will be sufficient since gamma distribution belongs to exponential family. In the Gamma distribution, $\sum X_{i}$ is complete and any function of this will also be complete. Hence, first two options are complete and sufficient. I have doubt in the completeness of third and fourth options. I know, they will be sufficient as contains the whole information of samples.
Best Answer
Yes, $\bar{X}$ is sufficient because you can write the joint density as $$ f(x_1,\ldots,x_n) = g[\bar{x},\theta]h(x_1,\ldots,x_n). $$
To see if $(X_1, \sum_{i=2}^n X_i)$ is sufficient, can you write the joint density as $$ \tilde{g}[(X_1, \sum_{i=2}^n X_i),\theta]h(x_1,\ldots,x_n)? $$
Regarding your other question proving completeness, try using linearity of the expectation operator. Is there some linear combination of the two statistics that has mean $0$, but isn't $0$ with probability $1$?