Prove $(\sum_{i=1}^{n}X_{i},\sum_{i=1}^{n}X_{i}^{2})$ is not a complete statistic for $N(\mu,\mu^2)$ distribution
Solution 1:
Consider the function $g$ of the sufficient statistic $T$ defined as $$g(T(X_1,\ldots,X_n))=2\left(\sum_{i=1}^n X_i\right)^2-(n+1)\sum_{i=1}^nX_i^2$$
For all possible values of $\mu$, verify that we have
$$E_{\mu}g(T(X_1,\ldots,X_n))=2n(1+n)\mu^2-(n+1)2n\mu^2=0$$
However, $g(T)$ is not identically zero with probability one.
So the minimal sufficient statistic $T$ is not complete.