Generalized likelihood ratio test

[BicycleTree]

This is the problem (t for theta):

X ~ Expo(t) = t * e ^ (-t * x), x>0, t >0
                          0 otherwise

Test H0: t <= 1 vs. H1: t >1 using the generalized likelihood ratio test where you have a random sample from X {X1, X2, ... , X50} and the sum of all Xi = 35. Use alpha = .05 (the probability of a type I error)

My professor actually did much of this in class and I've asked him and he said it's not supposed to be a hard problem. I don't know. In class, this is the test he arrived at for the GLRT:
reject H0 if

((t0 ^ n)* e^(-t0 * (sum over Xi))) / ((n/(sum over Xi))^n * e^(-n)) <= k

n is the number of observations (50), t0 is... hmm, I think t0 is 1 in this case. k is the number that makes the test agree with alpha = .05.

So now I need to simplify that expression so that I can find the probability that the resulting distribution is less than k. I can mess with k to make the left side simpler but I think I shouldn't do anything with (sum over Xi)--I think k should not depend on the values of the Xi's. I can only simplify it this far:
n * ln (sum over Xi) - t0 * (sum over Xi) <= k1

Do I need to use transformations to figure out how that messy thing on the left is distributed or am I doing something wrong? He said it shouldn't be that hard of a problem so I'm hesitant to use transformations.

 

[BicycleTree]

I did the uniformly most powerful test for that and got a reasonable answer (reject H0 if the sum over Xi <= 38.33). But the GLRT still escapes me... I'll try the transformation method for it later and see if that gets me somewhere.

 

[juvenal]

I'd like to take a look at this and try to answer your question, but I'm pretty swamped right now, so I'm not going to get to it anytime soon.