# Hypothesis Testing

• October 5th 2009, 09:52 PM
panda*
Hypothesis Testing
Let $X_{1}, ... , X_{n}$ be a random sample from an exponential distribution with mean $\theta$. Show that the likelihood ratio test of $H_{0} : \theta = \theta_{0}$ against $H_{1} : \theta = \theta_{1}$ has a critical region of the form $\sum^{n}_{i=1} x_{i} \leq c_{1}$ or $\sum^{n}_{i=1} x_{i} \geq c_{2}$. How would you modify this so that chi-square tables could be easily used?
• October 6th 2009, 08:16 AM
matheagle
The sum of indep EXP(theta's) is a $\Gamma(n,\theta)$

Call this sum of X_i's X and now transform to the chi square

A $\chi^2_a=\Gamma(a/2,2)$

So do some calculus one with the substitution $x/\theta=y/2$