Information Pertaining to the Final Exam

Basics

The exam period is 7:30-10:15 PM on Tuesday, Dec. 14. You are expected to take the exam during the official time slot. Exceptions to this policy will rarely be made.
(Note: The official time slot for the exam may be changed. For example, if there are too many class cancelations due to bad weather, or for any other reason, the Provost may alter the exam schedule. To be safe, it may be best to plan to remain in the area through Dec. 15, which is listed on GMU's calendar as a make-up day for exams.)

The exam is an open books and open notes exam. You can use whatever printed or written material that you bring with you to the exam. You cannot share books or notes during the exam.

You can use a calculator and/or computer during the exam if you wish to. (Some may wish to use software such as Maple or Mathematica.) However, you cannot connect to the internet, except to use Wolfram Alpha (or a similar online tool for calculus), the class Blackboard site, or an electronic copy of the text book. During the exam period you should not communicate in any way with another party while the exam is in progress.

Description of the Exam

The exam has six problems having a total of eleven parts. Each part is worth 10 points, and I'll count your best 10 of 11 problem scores to get an overall score out of 100 points. Altogether, 40 points pertain to point estimation, 20 points to interval estimation, and 50 points to hypothesis testing (and of the 7 parts on interval estimation and hypothesis testing, 3 of the parts pertain to asymptotic results and the other 4 parts pertain to exact methods).

The hypothesis testing methods that you should be comfortable with are most powerful tests based on the Neyman-Pearson lemma, UMP tests based on monotone likelihood ratios (noting that there are some special results that are useful for exponential family distributions), and (general) likelihood ratio tests.

Both confidence interval problems can be solved using pivots (and so for this exam you won't have to use the method of inverting the acceptance region of a test, or the method C&B refers to as "pivoting the cdf").

The point estimation methods covered are indicated in the description above. (Methods not referred to (e.g. Bayesian estimation and Pitman estimators) won't be covered on the exam.)
The exam will focus on basics: straightforward applications of widely-used methods in relatively simple settings. 3 of the 6 problems (and 6 of the 11 parts) pertain to making inferences about parameters of one-parameter exponential family distributions. If you understand the basics for each of the estimation and hypothesis testing methods I've indicated above, you shouldn't get bogged down in the calculus and probability as you apply them to the problems I've chosen for you (as long as you don't make things harder than they need to be). I'll be throwing you a few "softballs" --- but take your time and be careful with them ... in the past some students tend to struggle a lot with problems that are meant to be easy (the really simple settings throw them off more than help them I guess).

What to Study From the Course Notes

Since there were no homework problems based on the Chapter 10 material covered during the last class meeting, it'll be important to thoroughly study the appropriate portions of the courses notes. The portions of the Ch. 10 notes that are pertinent to the final exam are the first 4.5 pages of Sec. 10.3 and the first 4.5 pages of Sec. 10.4. (In the Sec. 10.3 notes, pay particular attention to the results pertaining to how to get approximate p-values from likelihood ratio tests and from UMP tests based on monotone likelihood ratio results for exponential familiy settings in which the test statistic can be viewed as a sum of iid random variables.)

For the rest of the course notes, a good way to review the most important parts is to focus on certain examples and then take detours when you realize that you need to review supporting material pertaining to things such as identifying sufficient statistics, obtaining CRLBs, and establishing monotone likelihood ratios. (Some special results pertaining to exponential families (e.g., results used on HW 1) could be useful, but are not absolutely necessary.)

Here's a list of good examples to review: With regard to point estimation methods, the exam will cover MMEs, MLEs, and UMVUEs, but not Bayesian and Pitman estimators. For hypothesis testing, know how to find the test function for a specified size test for MP tests based on the Neyman-Pearson lemma, UMP tests based on monotone likelihood ratios (both within and not within one-parameter exponential family setting), and likelihood ratio tests for cases in which a MP or UMP test doesn't exist. Know how to find the power of a test, and how to obtain a p-value. For confidence intervals, you should be comfortable with the methods covered up through p. 9.12 of the course notes. For the asymptotic material covered from Ch. 10, understand the important parts of the first five pages of the notes pertaining to the Ch. 10 coverage of both hypothesis tests and confidence intervals (so first five pages of each of these sections of the Ch. 10 notes). Obviously the exam won't cover all of these things, but what it does cover will be taken from this description.

What else? There won't be any problems like the problems on HW #1 and HW #2, or the Pitman estimator and Bayesian parts of HW #5, but some of what those assignments covered may come in handy in solving problems based on the topics of the paragraph above. Also, although I won't explicitly request a CRLB, for one of the problems you may want to obtain a CRLB.

Homework Problems to Review

I recommend trying to understand these homework problems, and not to worry about other homework problems this semester.
HW #3
1(a), 1(b), 2(a), 2(b), 5(a), 5(b)
HW #4
1(a), 1(c), 1(d), 2
HW #6
1, 2, 3(a), 3(b), 3(c), 4(a), 4(b), 5(a), 6
It might also be good to review the solutions for Quiz #3, Quiz #4, and Quiz #6.

The item in bold font above are the most important (and the Ch. 10 problems below are also very important).

Problems Based on Ch. 10 Material

(since none of the HW problems covered Ch. 10)
Problem A:
Consider 90 iid geometric random variables having parameter θ (and thus mean 1/θ), and a UMP test of the null hypothesis that θ <= 0.1 against the alternative that θ > 0.1. Give the approximate p-value (based on large sample results) which results for the case of the sum of the observations equal to 723.
answer:
0.025
Problem B:
Consider n iid random variables having the pdf indicated here. Give the approximate p-value (based on large sample results) which results from a LRT of the null hypothesis that θ = 1 against the general alternative for the case of n = 50 and the sum of the observations equal to 115.5.
answer:
0.14
Problem C:
Consider 100 independent random variables:
X1, X2, ..., X50
being iid normal random variables having mean 0 and variance θX, and
Y1, Y2, ..., Y50
being iid normal random variables having mean 0 and variance θY.
Given that
Σ xi2 = 37.5 & Σ yi2 = 62.5,
give the approximate p-value (based on large sample results) which results from a LRT of the null hypothesis that the two variances are equal against the general alternative (that the variances are not equal).
answer:
0.072
Problem D:
Consider 100 independent random variables:
X1, X2, ..., X50
being iid normal random variables having mean 0 and variance θX, and
Y1, Y2, ..., Y50
being iid normal random variables having mean 0 and variance θY.
Given that
Σ xi2 = 37.5 & Σ yi2 = 62.5,
give the approximate p-value (based on large sample results) which results from a LRT of the null hypothesis that the variances both equal 1 against the general alternative (that at least one of the two variances is not equal to 1).
answer:
0.20
Problem E:
Consider n iid random variables having the pdf indicated here. Give an approximate 95% confidence interval for θ (based on large sample results) for the case of n = 100 and the sum of the logs of the observations equal to -50.
answer:
(1.67, 2.49) if you do something similar to the example on p. 10.4.2 of the course notes (and (1.61, 2.39) if you use the standardized sum of the t(Xi) as an approximately standard normal pivot)
Problem F:
Consider n iid random variables having the pdf indicated here. Give an approximate 95% confidence interval for θ (based on large sample results) for the case of n = 100 and
Σ xi4 = 74.6.
answer:
(1.12, 1.67) if you do something similar to the example on p. 10.4.2 of the course notes, which is preferred