Statistics 630 – Overview of Mathematical Statistics

Sections 700 & 720, Spring 2012, Prof. D. Cline

Welcome!  This page announces and describes my course for the Spring 2012 semester.

This course is intended for graduate students in various fields who require an introduction to mathematical statistics.  It covers basic probability theory, including random variables and their distributions, and the theory of statistical inference from the likelihood point of view, including maximum likelihood estimation, confidence intervals, likelihood ratio tests and Bayesian methods.

The course will be taught online using lectures and notes prepared by Dr. Thomas Wehrly.  A weekly online question and answer session will be conducted by Dr. Daren Cline.

The R statistical programming software will be used throughout the course.  See instructions below to install.

Course lectures, homework assignments, etc. will be posted at the DoStat website, but you will turn in your homework and exams on WebAssign.   See instructions below to register.

For comments or questions, e-mail me (dcline@stat.tamu.edu) or contact the TAMU Statistics Department.

Homework Policy:

Homework assignments will be downloaded from DoStat. Homework solutions must be in a single PDF file and uploaded to WebAssign (not to DoStat).  You will need a scanner (or a fax-to-PDF service) for this.  See Homework and Exam Submissions for more information. You should be identified on the initial page with your TYPED name, course and section number. Your homework solutions must be your own work, not from outside sources, consistent with the university rules on academic integrity.  I expect you to follow this policy scrupulously.  Your performance on the exams is much more likely to be better. You may use: Your textbook and notes from class. Your notes, homework, etc., from a related class that you took or are taking. Discussion with the instructor or grader. Voluntary, mutual and cooperative discussion with other students currently taking the class.  There will be an online discussion board. Do not post complete solutions.  Suggestions, descriptions, partial explanations and final numerical answers are ok. You may not use: Solutions manuals (printed or electronic). Solutions, notes, homework, etc., from classes taught elsewhere or at another time. Solutions, notes, homework, etc., from students who took the class previously. Copying from students in this class, including expecting them to reveal complete solutions in "discussion".

Exam Policy:

Distance students must be proctored according to the directions you will be given.  See Homework and Exam Submissions for the online proctor application.  Even if you have assigned a proctor for previous classes, you may need to repeat the process. Exams will be downloaded from WebAssign (not from DoStat). Exam solutions must be scanned into a single PDF file only and uploaded to WebAssign.  There will be a limited time to do this.  You will need a scanner (or a fax-to-PDF service).  See Homework and Exam Submissions for more information. You should be identified on the initial page with your PRINTED name, course and section number. Your exam solutions must be your own work, consistent with the university rules on academic integrity. Each exam will be comprehensive and cumulative, and closed book/notes. I will not expect you to quote theorems and results explicitly but I do expect you to demonstrate that you can make use of them.  Specifically, you will need to: Show all your work.  This does not necessarily mean showing every individual algebraic or calculus step – but it must be clear what those steps would be. Identify (by number, name or description) any theorems or examples you use. Clearly identify the solution and/or the end of a proof or derivation. No other resources are acceptable (no calculator). Copies of old exams will be available for you to review.

Missed Work and Incompletes:

This is based on university policy. If you must miss an exam due to illness or circumstances beyond your control, notify me or the Statistics Department (before, if feasible, otherwise within two working days after you return).  See me as soon as possible to schedule a make-up exam. An Incomplete grade will be given only in the event that circumstances beyond your control cause prolonged absence from class and the work cannot be made up.

Students will have access to streaming videos of the lectures and Q&A sessions at the DoStat website.  Other resources and information for the course also will be accessible on the website.  To access the website, you will need to register according to the instructions below by the time classes start.  The course reference and registration code will be emailed to your TAMU account before the start of the semester. 

1. (If you are already registered as a DoStat user, skip steps 1 and 2 and go to step 3.)  Go to http://dostat.tamu.edu and click on the Register Here link.
2. Fill in all of the information and click on Submit.  Please enter your preferred email address as this will be the one I use to contact you.  You should get an email confirming your registration.
3. Login (at the same site) using the information entered in step 2.
4. Click on the Add Course link to the left.
5. Fill in the Course Reference DS-XXX and Registration Code XXXXXXX provided to you and press Register.
6. Click My Account and edit your information as desired.  Again, please indicate your preferred email address.

After registering for the course, take a look at the resources.  Please look/watch for further instructions concerning the use of DoStat and WebAssign, watching videos, attending the Q&A sessions, submission of homework and exams, and using R.

The recorded (streaming video) lectures will be found under "Lectures".  Under "Files" you will find printed lecture notes and examples that are presented in class.  Under "Links" there are several useful web links, including a link to the R statistical programming software that will be used to illustrate concepts throughout the course.  (You will need to install R.  See CRAN below.)

You will turn in your homework at the WebAssign website.  You will also download your exams and submit them at WebAssign.  We have changed the procedures for accessing WebAssign.  Detailed instructions, along with the CLASS KEY, will be posted on the DoStat instruction board. In short, they are as follows.

1. Go to https://www.webassign.net.
2. Click on I HAVE A CLASS KEY and enter the key for this course (which I will send before classes start and will be posted on DoStat).
3. Verify the course number and instructor, and click YES if it is correct.  Otherwise, click NO and repeat step 2 with the correct key.
4. If you do not have a WebAssign account, select the option to create an account.  If you already have a WebAssign account, select the appropriate option and enter your login information.  Click Continue.
5. Select the My Assignments tab.  This is where you will submit homework and exams and download exams.  See the examples provided.  Verify that you are able to download and upload PDF files, using the examples.

Homework and exams must be submitted as single PDF files no larger than 15MB.  You will need a scanner for this.  You may submit homework multiple times as long as it is before the due date/time.  Only the last submission will be graded.  There will be a limited time frame in which you can download, take and submit exams.

All students will need to download and install the latest R software.  R is a statistical programming language we will use for simulation, computing probabilities and trying out ideas.  It may be obtained at the CRAN website, as described below.

1. It is recommended that you first uninstall previous versions of R, if you have any.
2. Go to http://lib.stat.cmu.edu/R/CRAN and click your choice of platform (Linux, MacOS X or Windows) for the precompiled binary distribution.  Note the FAQs link to the left for additional information.
3. Follow the instructions for installing the base system software (which is all you will need).

Examples using R, that you can mimic, will be given in the lectures.

The Question & Answer sessions will be conducted live online each Wednesday evening, using the TTVN Centra service.  We will give you the specific class link prior to the start of classes.  Meanwhile, we recommend that you run the System Check at http://webconference.tamus.edu/main/COS/stat.   (Click on the Centra System Check link under the Statistics logo at the top of the page.)

I often am asked about the required calculus for this course, especially by students who took it years ago (and haven't used it since!).

The courses listed as prerequisites are the 1st–3rd semester calculus courses at TAMU, which cover through multi-variable calculus and are pretty standard.  This includes derivatives and integrals for power, logarithm and exponential functions, use of the chain rule and integration by parts, as well as double integrals.  It also includes being able to understand and compute limits.  You can refresh your knowledge with some online links I will provide in DoStat, for example.  There will be questions on exams that involve derivatives and integrals.  However, I keep it fairly simple and there are many examples in the book and notes.  You also will learn to recognize a few definite integrals.  We mainly want you to understand the principles involved – the most important integrals have to be computed numerically anyway and are included in any decent statistical software.

We also will be making extensive use of the (natural) logarithm and exponential functions, so it is a good idea to review them as well.

Many students tend to have a problem keeping the variables straight – and we usually will be dealing with several simultaneously.  They integrate over "x" when it is "y" they need to integrate over, or they compute a definite integral over "x" and end up with something that still depends on "x".  Things like that.  Try to get comfortable using variables named with letters other than "x".  "x" is just a name anyway; it does not affect how you do integrals or derivatives.

In addition, we will have variables with different roles: for the arguments of functions, for random variables and for parameters.  While we can use notational devices to help distinguish them, it will not always be clear.  The more you endeavor to pay attention to the role of each variable, the less confusing it will be.

Topic (order varies in the lecture notes)		Textbook Section
1. Introduction to Probability
	A. Interpretation, experiments, sample space, events, set theory	1.1–1.2
	B. Definition and properties of probability models	1.2–1.3
	C. Finite sample spaces, counting methods, combinatorial methods	1.4
	D. Conditional probability, independence, Bayes' theorem	1.5
2. Random Variables and Distributions
	A. Random variables and distributions	2.1–2.2
	B. Discrete random variables, continuous random variables	2.3–2.4
	C. Cumulative distribution functions	2.5
	D. Functions of a random variable	2.6
	E. Joint distributions, conditional distributions, independence	2.7–2.8
	F. Functions of multiple random variables	2.9
	G. Simulating random variables	2.10
3. Expected Values
	A. Expectation of discrete and continuous random variables	3.1–3.2
	B. Variance, covariance and correlation	3.3
	C. Moments, generating functions	3.4
	D. Conditional expectations	3.5
	E. Inequalities for probability and expectation	3.6
4. Sampling Distributions and Limit Theorems
	A. Sampling distribution of a statistic	4.1
	B. Convergence in probability and the weak law of large numbers	4.2
	C. Convergence in distribution and the central limit theorem	4.4
	D. Monte Carlo approximations	4.5
	E. Normal random samples and related distributions	4.6
5. Introduction to Statistical Inference
	A. Statistical models	5.1–5.3
	B. Data collection and summary, types of statistical inference	5.4–5.5
6. Likelihood Inference
	A. Likelihood function and maximum likelihood estimation	6.1–6.2
	B. Bias, variance, mean squared error	6.3.1
	C. Confidence intervals, construction using pivots	6.3.2
	D. Hypothesis testing and the construction of tests	6.3.3
	E. Method of moments estimators, the bootstrap	6.4
	F. Large sample properties of maximum likelihood estimators	6.5
	G. Large sample approximate confidence intervals	6.5
7. Bayesian Inference
	A. Prior and posterior distributions	7.1
	B. Inferences based on the posterior distribution	7.2
8. Testing Hypotheses
	A. Neyman-Pearson approach to hypothesis testing	8.2.1–8.2.4
	B. Generalized likelihood ratio tests, applications	8.2.5
	C. Wald and score tests

Copyright Information

Each document provided on these web pages is copyrighted by me (Daren B.H. Cline) with all rights reserved, whether or not the document explicitly states so.  These documents may only be used for academic purposes and they may not be reproduced or sold without my permission.  That means you may refer to them for other classes or for research, just as you would any book, as long as neither you nor anyone else reproduces them for sale or other distribution.  If you would like to use some of the material for instruction, you need to first get written permission from me (Daren B.H. Cline, TAMU Department of Statistics, College Station TX 77845-3143).