
Hypothesis Testing
Let $\displaystyle X_{1}, ... , X_{n}$ be a random sample from an exponential distribution with mean $\displaystyle \theta$. Show that the likelihood ratio test of $\displaystyle H_{0} : \theta = \theta_{0}$ against $\displaystyle H_{1} : \theta = \theta_{1}$ has a critical region of the form $\displaystyle \sum^{n}_{i=1} x_{i} \leq c_{1}$ or $\displaystyle \sum^{n}_{i=1} x_{i} \geq c_{2}$. How would you modify this so that chisquare tables could be easily used?

The sum of indep EXP(theta's) is a $\displaystyle \Gamma(n,\theta)$
Call this sum of X_i's X and now transform to the chi square
A $\displaystyle \chi^2_a=\Gamma(a/2,2)$
So do some calculus one with the substitution $\displaystyle x/\theta=y/2$