Could you please take a look if my answer makes sense- I don't have a solution to this practice question. Also, I am not quite sure how to read the data in part (c).

Question

Let be a random sample from an exponential distribution with density function

where is an unknown parameter.

(a) Obtain the likelihood ratio test statistic for testing against .

(b) State the assymptotic distribution of twice the logarithm of the likelihood ratio test statistic under Ho.

(c) If , will you reject the null hypothesis Ho at a significance level of 5%?

Answer.

(a)

the MLE for is (I checked against a few sources).

Therefore my likelihood ratio

(b) the assymptotic distribution

(coverges in distribution)

(c) calculate r from the given test results.

Now I am not sure what is "Y" - if that is then why the MLE realisation ( ) is so far away from ?

Ideally I would use (b) to test the hypothesis.

Does the part (c) data make sense to you?