Show that the statistic $T(x_1,...,x_n)=\sum_{i=1}^nx_i^2$ is complete.

Question

Let $X_1,...,X_n$ be iid random sample form $N(\theta,c\theta)$, where $c$ is a known constant. Show that the statistic $T(x_1,...,x_n)=\sum_{i=1}^nx_i^2$ is complete.

In other words I have to show that if $$E[g(T)]=0\quad \Longrightarrow \quad g(T)=0 \quad \forall t$$

I'm kind of stuck with the distribution of $T$ because $x_i^2$ isn´t a gaussian distribution. Any suggestions on how to approach this problem would be great!

Answer 1

This is Proposition 2.1 on page 110 in [Jun Shao]Mathematical Statistics(2nd edition).