Consider a random sample of size $n$ from a two-parameter exponential distribution, $X_i \sim $EXP($\theta,\eta$), and let $\eta^*$ and $\theta^*$ be the MLEs.
a) Show that $\eta^*$ and $\theta^*$ are independent.
b) Let $V_1=2n(\overline{X}-\eta)/\theta$, $V_2=2n(\eta^*-eta)/\theta$, and $V_3=2n\theta^*/\theta$. Show that $V_1 \sim \chi^2(2n)$, $V_2 \sim \chi^2(2)$, and $V_3 \sim \chi^2(2n-2)$. Hint: Note that $V_1=V_2+V_3$ and that $V_2$ and $V_3$ are independent. Find the MGF of $V_3$.
So I'm not really sure how to do part a) at all, and for part b), I know I can just show $V_2$ and $V_3$ and then add them to get $V_1$, but I'm not really sure how to get those 2. I'm assuming you somehow use the rule that if $X \sim \Gamma(\alpha,\beta)$ then $2X/\beta \sim \chi^2(2\alpha)$ but I'm not sure how to apply that.
