Let $(x_{1}, \ldots, x_{n})$ be a sample from some known distribution $F_{\theta}$ with unknown parameter $\theta$ and let $$\mathbb{E}_{\theta}{X_{i}} = \theta$$ Let $T$ be an identity function of a sample e.g. for the given sample $T$ returns the sample itself. How to prove that $T$ is not a complete statistics?
By definition, the complete statistics is a function $T$ such that for any measurable $g$ such that $$\mathbb{E}(g(T)) = 0$$ for any $\theta$ $$\mathbb{P}_{\theta}(g(T) = 0) = 1$$ for any $\theta$.
I guess that the problem is quite easy but i cannot find an easy way to attack it. Are there any hints that might help?
Take $g(T)=X_1-X_2$. Then $E(g(T))=0$ but $g$ is not identically $0$ with probability $1$.