See the attachment for the fully formatted solution.
1. Let -- be a random sample from a uniform distribution on --.
a) Find the MLE --.
b) Show that the distribution of -- does not depend on -- and use it to set 1 - alpha confidence interval for --.
c) Show that the likelihood rate test statistic -- for testing --- against ---.
2. Let -- be a random sample from a Bernoulli distribution with pmf --, zero elsewhere, where --
a) We wish to test ---. Find -- and -- where -- is the likelihood ratio test statistics.
b) Determine the Wald-type test and give, explicitly, the rejectiomn rule for a level alpha test for Part a).
c) Determine the limiting distribution.
The solution illustrates the determination of MLE and likelihood ratio test statistic for a given problem. The response includes a Word and Excel attachment.