AMS570.01 Midterm Exam 2 Spring 2018

Name: ______ID: ______Signature: ______

Instruction: This is a close book exam. You are allowed a one-page 8x11 formula sheet (2-sided). No cellphone or calculator or computer is allowed. Cheating shall result in a course grade of F. Please provide complete solutions for full credit. The exam goes from 8:30-9:50am. Good luck!

1.Given a random sample from a Normal population with mean and variance 1. Please

(a)Derive the maximum likelihood estimator (MLE) of.

(b)Assuming the prior ofDerive the the Bayes estimator of.

(c)Which of the two estimators (the Bayes estimator and the MLE) are better? Why?

(d) Derive the 100(1-α)% Bayesian HPD credible set for .

Solution:

(a) For MLE,

Let

We have .

(b) The posterior distribution of ,

So follow, Bayes estimator is the mean of the posterior distribution, so we have

, ,

Once we see that in general, the Bayes estimator is biased and could have larger MSE than the MLE. However, when the prior information is accurate, for example,taking the extreme case of. Inthis case, the Bayes estimator is not only unbiased, but also has smaller MSE than the MLE. Therefore, when we do have reliable prior information, the Bayesian estimator is preferred.

(d) Since,

We know that =  , which we can plug into the first equation and solve for in the middle to get

2. Let and be two independent samples. Furthermore, σ2 is unknown. Please derive the best100(1-α)% confidence interval for

Our point estimator is =

So we have

So our pivotal quantity is

Then, by calculating T,

we have

So the best 100(1-α)% confidence interval for is [].

3.We have two independent samples and where σ2 is known. For the hypothesis of

(a) At the significance level α, please derive the most powerful test for the above hypotheses.

(b) Please derive the general formula for power calculation for the above test based on an effect size of EFF at the significance level of α.

Definition: Effect size = EFF =|| (e.g. Eff=1)

(c)With a sample size of 50 per group, α = 0.025, and an estimated effect size of 1, please calculate the power of your test.

Solution:

(a) From the question, we could easily get

Since it contains twoparameters, we take the partial derivatives with µ and set the partial derivatives equal to 0. Solving it we have:

Also, we have, , so we have where .

We take the partial derivatives with , and set the partial derivatives equal to 0. Solving it wehave:

Then, we have,

Expand

By simplifying this equation, we could have

Thus, we have

According to the Neyman-Pearson Lemma, the likelihood ratio test for a simple nullhypothesisversusasimple alternativehypothesis is a most powerful test. Thus we have shown that the most powerful test will reject when .

(b)

(c)