Generalized likelihood ratio test

In summary, my professor believes that the problem with the generalized likelihood ratio test is that you shouldn't depend on the values of the Xi's. Instead, you should use a transformation to simplify the expression on the left side of the equation. If you use a transformation, you can get a reasonable answer to the question of whether or not H0 is rejected.
  • #1
BicycleTree
520
0
This is the problem (t for theta):
Code:
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.
 
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  • #2
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.
 
  • #3
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.
 

Related to Generalized likelihood ratio test

1. What is a Generalized Likelihood Ratio Test (GLRT)?

The Generalized Likelihood Ratio Test (GLRT) is a statistical hypothesis test used to compare two statistical models. It is used to determine whether one model is significantly better at explaining the data than the other.

2. When is a GLRT used?

A GLRT is used when the data being analyzed follows a specific distribution (such as a normal, binomial, or Poisson distribution) and when the parameters of the distribution are unknown. It is commonly used in fields such as economics, psychology, and biology.

3. How does a GLRT work?

A GLRT works by calculating the likelihood function for each model and then taking the ratio of the likelihood functions. This ratio is then compared to a critical value, typically obtained from a chi-square distribution, to determine whether the model with the higher likelihood is significantly better.

4. What are the assumptions of a GLRT?

The main assumption of a GLRT is that the data being analyzed follows a known distribution, and that the parameters of the distribution are unknown. Additionally, the data should be independent and identically distributed.

5. What are the advantages of using a GLRT?

One of the main advantages of using a GLRT is that it is a relatively simple and straightforward method for comparing two models. It also does not require any assumptions about the underlying distribution of the data, making it a more flexible approach. Additionally, it is a powerful tool for detecting small differences between models.

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