# Generalized likelihood ratio test

• BicycleTree
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.
BicycleTree
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.

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.

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.

## 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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