Department of Mathematics and Statistics, Binghamton University people:qiao:teach

(Archive) Math 448 Introduction to Probability and Statistics II.

2016-01-24T18:42:22-04:00

Math 448 Introduction to Probability and Statistics II.
Fall 2014

Instructor: Xingye Qiao
Email: qiao@math.binghamton.edu
Phone number: (607) 777-2593
Office: OW-134
Meeting time & location: MWF 3:30-5:00 at SW 321.
Office hours: T 3:00-5:00, F 11:00-12:00
If you need to reach me, please e-mail qiao@math.binghamton.edu.
Please include [Math448] in the subject line of your email, or your email may not be read promptly.

Prerequisite

Math 447 (grade C or above). Probability is the foundation of statistics. There will be a probability aptitude test (referred to as the PAT below) scheduled on Sept. 15. Please review the materials in Math 447 as early and as thorough as possible. Failure to display adequate aptitude in probability may lead to difficulty in the current course.

Learning Objectives

Point estimation, confidence intervals and hypothesis testing.
Linear regression model.
Basic statistical software - R. Data input and manipulation. Plots. Model and formula. Simulation.

This course is a 4-credit course, which means that students are expected to do at least 12.5 hours of course-related work or activity each week during the semester. This includes scheduled class lecture/discussion meeting times as well as time spent completing assigned readings, studying for tests and examinations, preparing written and computing assignments, and other course-related tasks.

Required Textbook

Mathematical Statistics with Applications (7th ed.) by Wackerly, Mendenhall, and Scheaffer.

This is the course text. All homework assignments will come from this book.
A customized soft cover version used at BU is available from the University Bookstore. The custom book and the hardcover book are equivalent for the purpose of this course except that the former is offered at an affordable price, while the complete version may have a higher resell value. Students may choose whichever one to purchase.

In the University Bookstore, the list price for the soft cover version is \$167, and for the hard cover version \$315 (as of 2014). Amazon has the hard cover book at price of $247. You may also try to rent the textbook from providers such as Amazon.

Online resources for R

Downloads:

R - mirror hosted at UC Berkeley.
R Studio - a more user friendly platform for R.

Class Attendance

Attendance is partially mandatory, enforced by the daily quizzes. Following the academic policy listed in the University Bulletin, the instructor will NOT grade exams of any student missing more than 25% of the quizzes. The final grade will be an F if a student misses more than 25% if the quizzes. See more details in the Grading section below.

For the semester of Fall 2014, missing more than 8 quizzes without an advance notice will lead to an F.

Grading

Quizzes (including PAT) (32 %), three tests (45 %, with 15% each) and a final exam (23 %).
There are 31 regular quizzes scheduled, one in each class session. No quiz is scheduled for a class session that is immediate after a test. Each quiz will be graded on the scale from 0 to 10.
The Probability Aptitude Test (PAT) scheduled on Sept. 15 should be treated as a special “large” quiz and will be graded on the scale from 0 to 40.
If you miss an exam, test or quiz, your score for that exam, test or quiz will be a zero.
The lowest three regular quiz grades will be dropped when the final total grade is calculated. Hence only 28 out of the 31 quizzes are counted.
In addition, three waivers can be granted for absences from quiz only with an advance notice. You may use them for interviews, sickness, or personal affairs. No proof is needed to request a waiver. You may include expressions such as [waiver], [absence] or [immunity] in the subject of the email so that your request can be noted properly.
Bonus points: If a student finishes all the quizzes (after excluding up to 3 waivers), 15 bonus points will be added to his/her grade.
Bonus points: If a student finishes all but one quiz (after excluding up to 3 waivers), 5 bonus points will be added to his/her grade.
Waivers are used for the sole purpose of determining the eligibility of the bonus points. Please distinguish it from the ”dropping the three lowest grades” rule. For examples,
1. Student A has missed three quizzes but she has requested three waivers. As she has finished all but the three quizzes that she requested waivers, she will get the 15 bonus points. The grades for the quizzes that she missed will be zero. But since the three lowest quiz grades are dropped, her grade will not be negatively affected.
2. Student B has two quizzes missing and he did not request any waiver. The two zeros will be dropped anyway, along with the lower grades for the quizzes that he took. But he will not receive the 15 or 5 bonus points.
3. Student C has seven quizzes missing and she requested three waivers. She will not receive bonus points. Three of the zeros will be dropped. But there are still four zeros what will go to her final grade. In this case, whether she requested waivers at all does not make any practical difference since either way she cannot get the bonus points.
4. Student D has missed 12 quizzes. Whether or not he has requested waivers for these absences, he will receive an F in the course because he missed more than 25% of the quizzes.

Components	Dates	Points	Time allowed
Quiz	Daily	280	10 minutes/day * (31-3) days
Probability Aptitude Test (PAT)	Sept. 15	40	40 minutes
Test 1	Sept. 29	150	90 minutes
Test 2	Oct. 24	150	90 minutes
Test 3	Nov. 17	150	90 minutes
Exam	Dec. 15	230	120 minutes
TOTAL		1000

Dates for Quizzes, Tests and Exam

Monday	Wednesday	Friday
	9/3	9/5
9/8	9/10	9/12
9/15 PAT	9/17	9/19
9/22	~~9/24~~	~~9/26~~
9/29 Test 1	10/1	~~10/3~~
10/6	10/8	10/10
10/13	10/15	10/17
10/20	10/22	10/24 Test 2
10/27	10/29	10/31
11/3	11/5	11/7
11/10	11/12	11/14
11/17 Test 3	11/19	11/21
11/24	~~11/26~~	~~11/28~~
12/1	12/3	12/5
12/8	12/10	12/12
12/15 final exam 12:50-14:50 at SW321

Regular quizzes: dates in bold-face.
PAT, Tests & Exam: dates underscored.
No class: dates stroke-through.

Homework Assignment

Homework will be assigned after each class session and be posted at http://blackboard.binghamton.edu/.
Homework assignments will not be collected. Students are welcome to discuss the homework with the instructor during office hours. There is a solution manual on the market which provides detailed solutions to half of the questions.

Quiz

Quizzes will be given at the end of a class session.
Quiz problems are chosen from previous homework assignments either in exactly same forms or with some modifications. It is highly recommended that a student finishes homework by him- or herself.
Quizzes are always closed-booked.
No make-up is given for quizzes.

Some Deadlines

Sept. 12: Course add and drop/delete deadline.
Oct. 31: Course withdraw/change grade option deadline.

Note that a “Pass” grade in the “Pass/Fail” grade option does not count toward math degrees. If you are a math major, it is not advised to change the grade option to “Pass/Fail” unless you are ready to retake the course at a later time.

Make-ups

If you need to take a make-up, if possible, an advance request should be given. Checkable written proof to justify the request should be given.

In order to minimize the need for make-up exams and the stress of dealing with multiple exams, within the first two weeks of the semester, all students must check the exam schedules of other courses they are taking and make sure that there is no major conflict. The exam dates may be changed accordingly only if the instructor determines necessary.

Academic Dishonesty

Students found cheating will be reported to the Provost Office following the academic procedure listed in the University Bulletin. Laptop and electrical communication devices cannot be used in a quiz, test or exam. Calculator in a cellphone cannot be used. Calculators are in general not allowed.

Calculators are in general not allowed. If you are used to using calculators, you should practice on homework problems without using a calculator.

Disciplines

No laptop usage in classroom. Text messaging should be minimal. Late arrivals, early departures, cell phone conversations, eating and drinking, etc., are inappropriate behaviors. According to the Faculty-Staff Handbook, the instructor may ask those who, in the instructor’s judgment, have seriously impaired the class’s ability to achieve the objectiveness of the course, to leave the classroom.

How to succeed in this course

Read the book once, before class!
Do not try to study by reading the book many times. Come to the class and listen to the lecture. Be proactive in class. Ask 'why?'. Focus on the motivations.
Don't solve a question by matching it to a formula in your memory. To understand statistical procedures is much easier than to memorize (and search for) these formulas.
Do practice more on probability skills. You need them.
Don't be ashamed for low quiz grades. To be challenged is part of the life and is a very good way of study.
Do not skip class! You may never be able to make it up. The nature of the course decides that materials are built one upon another.
This is not a “read the book three times the night before the exam ⇒ get an A” class.

(Archive) Math 448 Mathematical Statistics.

2016-01-24T18:42:52-04:00

Math 448 Mathematical Statistics.
Fall 2015

	Section 01	Section 02
Instructor:	Xingye Qiao	Ganggang Xu
Email:	qiao@math.binghamton.edu	gang@math.binghamton.edu
Phone number:	(607) 777-2593	(607) 777-3550
Office:	WH-134	WH-133
Meeting time:	MWF 8:00–9:30
Location:	LH 003	FA 246
Office hours:	MW 3:00–4:00 F 10:00–11:00	MW 3:30–5:00

Please include [Math448] in the subject line of your email, or your email may not be read promptly.

Prerequisites

Math 447 with a grade of C or better. Probability is the foundation of developing statistical inference. There will be a probability aptitude test (referred to as the PAT below) at the beginning of the course. Please review materials in Math 447 as early and as thoroughly as possible, especially if you took Math 447 semesters ago. Lack of aptitude in probability may increase the difficulty in the current course.

Learning Objectives

Understand the fundamental idea of statistical inference; conduct standard inferences including point estimation, confidence interval and hypothesis testing.
Derive, evaluate and compare point estimators and confidence intervals. Apply statistical inference to simple linear regression models.
Use statistical software (R) to manipulate data, conduct simple statistical inferences, conduct simple linear regression analyses, simulate data, etc.

Required Textbook

Mathematical Statistics with Applications (7th ed.) by Wackerly, Mendenhall, and Scheaffer.

This is the course text. Most if not all homework assignments will come from this book.
A customized soft cover version used at BU is available from the University Bookstore. The custom book and the hardcover book are equivalent for the purpose of this course except that the former is offered at an affordable price, while the complete version may have a higher resell value. Students may choose whichever one to purchase.

In the University Bookstore, the list price for the soft cover version is 167 USD, and for the hard cover version 315 USD (as of 2014). Amazon has the hard cover book at price of 247 USD. You may also try to rent the textbook from providers such as Amazon.

Online resources for R

R is chosen to be the statistical software used in the current course. There are many online resources where the students can learn the basics of R.

Please install R before the beginning of the semester. In addition to R, some may find RStudio to be handy. Downloads:

R - mirror hosted at UC Berkeley.
R Studio - a more user friendly platform for R.

Computing Homework

Grading

Components	Dates	Points	Time allowed
Quiz & homework	Daily	200	Quiz: 10-20 minutes/day if given
Probability Aptitude Test (PAT)	Sep 09	40	40 minutes
Software assignments		40
Test 1	Oct 2	160	90 minutes
Test 2	Oct 26	160	90 minutes
Test 3	Nov 23	160	90 minutes
Exam	TBD	240	120 minutes
TOTAL		1000
* Attendance Bonus		5–20
* Missing more than 12 classes (including three waivers)		F

Quiz and homework

At the end of each class session, the instructor can choose to administer a quiz or to collect the homework assigned at the last class session.

All problems in the quiz will be graded.
When homework is collected, the instructor will check whether all the problems have been finished. The instructor may choose to randomly grade a subset of problems for the whole or a subset of the student body.
A score between 1 to 10 will be given to each quiz or collected homework in each class session.
The lowest three scores among all scores over the semester will be dropped when the final grade for the quiz/homework component is calculated.
All the scores will be posted at the blackboard.
No make-up shall be arranged for quizzes.
A student who did not take the quiz or submitted the homework will receive 0 for the date. This is counted as an absence. See Class Attendance below.

Attendance Bonus

Attendance is partially mandatory, enforced by the daily quiz/homework. Full attendance will be rewarded as follows:

Up to three waivers can be granted only if the student gives advance notice. The student needs to send an email to the instructor with the date that an absence is expected. No reason or proof is needed. The waivers are intended for the students to attend job interviews and other matters.
20 bonus points will be added if the student has attended all the class sessions (except those that are waived).
10 bonus points will be added if the student has attended all but one class sessions (except those that are waived).
5 bonus points will be added if the student has attended all but two class sessions (except those that are waived).

Class Attendance

Following the academic policy listed in the University Bulletin and the faculty-staff handbook, the instructor will NOT grade exams of any student missing more than 25% of the quiz/homework. The final grade will be an F if a student misses more than 25% of the quiz/homework. In particular, faculty-staff handbook, VII.B.2. stipulates that

instructors have the right to deny a student the privilege of taking the final examination or of receiving credit for the course, or may prescribe other academic penalties if the student misses more than 25 percent of the total class sessions. Excessive tardiness may count as absence.

For Fall 2015, missing 12 classes will lead to an F.

Calendar

Week	Monday	Wednesday	Friday
1	Aug-31	Sep-02	Sep-04
2	~~Sep-07~~	Sep-09: PAT	Sep-11
3	~~Sep-14~~	Sep-16	Sep-18
4	Sep-21	~~Sep-23~~	Sep-25
5	Sep-28	Sep-30	Oct-02: Test 1
6	Oct-05	Oct-07	Oct-09
7	Oct-12	Oct-14	Oct-16
8	Oct-19	Oct-21	Oct-23
9	Oct-26	Oct-28: Test 2	Oct-30
10	Nov-02	Nov-04	Nov-06
11	Nov-09	Nov-11	Nov-13
12	Nov-16	Nov-18	Nov-20
13	Nov-23: Test 3	Nov-25	~~Nov-27~~
14	Nov-30	Dec-02	Dec-04
15	Dec-07	Dec-09	Dec-11
16	Dec-14

42 class sessions (39 regular sessions + 3 full exam sessions) * 1.5 hours $=$ 63 hours.

Sep 11: Course add and drop/delete deadline.
Oct 30: Course withdraw/change grade option deadline.

Make-ups

If you need to take a make-up, if possible, an advance request should be given. Checkable written proof to justify the request should be given.

Academic Dishonesty

Calculators are allowed for quizzes and tests.

Disciplines

Text messaging should be minimal. Late arrivals, early departures, cell phone conversations, eating and drinking, etc., are inappropriate behaviors. According to the Faculty-Staff Handbook, the instructor may ask those who, in the instructor’s judgment, have seriously impaired the class’s ability to achieve the objectiveness of the course, to leave the classroom.

How to succeed in this course

Read the book once, before class!
Do not try to study by reading the book many times. Come to the class and listen to the lecture. Be proactive in class. Ask 'why?'. Focus on the motivations.
Don't solve a question by matching it to a formula in your memory. To understand statistical procedures is much easier than to memorize (and search for) these formulas.
Practice more on probability skills. You need them.
Don't be ashamed for low quiz grades. To be challenged is part of the life and is a very good way of study. Too many easy materials make people boring.
Do not skip class! You may never be able to make it up. The nature of the course decides that materials are built one upon another.
This is not a “read the book three times the night before the exam ⇒ get an A” class.

(Archive) Math 448 Mathematical Statistics.

2017-01-10T20:12:28-04:00

Math 448 Mathematical Statistics.
Spring 2016

	Section 01
Instructor:	Xingye Qiao
Email:	qiao@math.binghamton.edu
Phone number:	(607) 777-2593
Office:	WH-134
Meeting time:	MWF 8:00–9:30
Location:	S2-143
Office hours:	MW 3–4 and F 9:45–10:45
Grader:	Changwei Zhou
Grader's office hour:	Send email to him (zhou@math.binghamton.edu) to make appointments.

Please include [Math448] in the subject line of your email, or your email may not be read promptly. In principle, only those questions regarding grading issues shall be addressed to the grader. Other questions should be addressed to the instructor.

Prerequisites

Learning Objectives

Understand the fundamental idea of statistical inference; conduct standard inferences including point estimation, confidence interval and hypothesis testing.
Derive, evaluate and compare point estimators and confidence intervals. Apply statistical inference to simple linear regression models.
Use statistical software (R) to manipulate data, conduct simple statistical inferences, conduct simple linear regression analyses, simulate data, etc.

Required Textbook

Mathematical Statistics with Applications (7th ed.) by Wackerly, Mendenhall, and Scheaffer.

This is the course text. Most if not all homework assignments will come from this book.
A customized soft cover version used at BU is available from the University Bookstore. The custom book and the hardcover book are equivalent for the purpose of this course except that the former is offered at an affordable price, while the complete version may have a higher resell value. Students may choose whichever one to purchase.

Online resources for R

R is chosen to be the statistical software used in the current course. There are many online resources where the students can learn the basics of R.

Please install R before the beginning of the semester. In addition to R, some may find RStudio to be handy. Downloads:

R - mirror hosted at UC Berkeley.
R Studio - a more user friendly platform for R.

Computing Homework

Grading

Components	Dates	Points	Time allowed
Quiz, homework & computing assignments	Daily	200	Quiz: 10-20 minutes/day if given
Probability Aptitude Test (PAT)	Sep 09	20	40 minutes
Test 1	Feb 17	180	90 minutes
Test 2	Mar 18	180	90 minutes
Test 3	Apr 25	180	90 minutes
Exam	TBD	240	120 minutes
TOTAL		1000
* Attendance Bonus		5–20
* Missing more than 12 classes (including three waivers)		F

Quiz, homework & computing assignments

At the end of each class session, the instructor chooses to administer a quiz or to collect the written homework assigned at the last class session.

All problems in the quiz will be graded.
Normally, if a quiz is administered, then the written homework will not be collected.
When the written homework is collected, the grader will check whether all the problems have been duly finished by the student. The grader may choose to randomly grade a subset of problems.
On each lecture day, a score between 1 to 10 will be given to the quiz given on that day or written homework collected that day. Counter-intuitively, the score 1 indicates a quiz or homework with no work / effort shown on paper or completely incorrect answers. The 1 point is credited simply because the student attends the class.
A score 0 will be recorded if the student did not render either a quiz or written homework. In this case, the zero score indicates that the students did not show up in the class on that day. This will have some serious punitive consequence (see Class Attendance below..)
In addition to the quiz/written homework on each day, there will also be some computing homework assignments. Submission of the work is done by an online form. A score between 0-10 will be given to each computing homework. There will be about 5 to 10 such assignments.
The lowest five scores among all scores above over the semester (including those for the quizzes, written homework and computing homework) will be dropped when the final grade is calculated.
All the scores will be posted at the blackboard.
No make-up shall be arranged for quizzes.

Attendance Bonus

Attendance is partially mandatory, enforced by the daily quiz/homework. Full attendance will be rewarded as follows:

Up to three waivers can be granted only if the student gives advance notice. The student needs to send an email to the instructor with the date that an absence is expected. No reason or proof is needed. The waivers are intended for the students to attend job interviews and other matters.
20 bonus points will be added if the student has attended all the class sessions (except those that are waived).
10 bonus points will be added if the student has attended all but one class sessions (except those that are waived).
5 bonus points will be added if the student has attended all but two class sessions (except those that are waived).

Class Attendance

instructors have the right to deny a student the privilege of taking the final examination or of receiving credit for the course, or may prescribe other academic penalties if the student misses more than 25 percent of the total class sessions. Excessive tardiness may count as absence.

For Spring 2016, missing 12 classes or more will lead to an F.

Calendar

Week	Monday	Wednesday	Friday
1	Jan-25	Jan-27	Jan-29
2	Feb-1	Feb-3: PAT	Feb-5
3	Feb-8	Feb-10	Feb-12
4	Feb-15	Feb-17: Test 1	Feb-19
5	Feb-22	Feb-24	Feb-26
6	Feb-29	Mar-2	Mar-4
7	Mar-7	Mar-9	Mar-11
8	Mar-14	Mar-16	Mar-18: Test 2
9	Mar-21	Mar-23	Mar-25
10	~~Mar-28~~	~~Mar-30~~	~~Apr-1~~
11	Apr-4	Apr-6	Apr-8
12	Apr-11	Apr-13	Apr-15
13	Apr-18	Apr-20	Apr-22
14	Apr-25: Test 3	Apr-27	Apr-29
15	May-2	May-4	May-6
16	May-9	May-11

44 class sessions (41 regular sessions + 3 full exam sessions) * 1.5 hours $=$ 66 hours.

Feb 5: Course add and drop/delete deadline.
Mar 24: Course withdraw/change grade option deadline.

Make-ups

If you need to take a make-up for an exam or test, if possible, an advance request should be given. Checkable written proof to justify the request should be given.

Academic Dishonesty

Calculators are allowed for quizzes and tests.

Disciplines

How to succeed in this course

Read the book once, before class!
Do not try to study by reading the book many times. Come to the class and listen to the lecture. Be proactive in class. Ask 'why?'. Focus on the motivations.
Don't solve a question by matching it to a formula in your memory. To understand statistical procedures is much easier than to memorize (and search for) these formulas.
Practice more on probability skills. You need them.
Don't be ashamed for low quiz grades. To be challenged is part of the life and is a very good way of study. Too many easy materials make people boring.
Do not skip class! You may never be able to make it up. The nature of the course decides that materials are built one upon another.
This is not a “read the book three times the night before the exam and you will get an A” class.

(Archive) Math 502 Statistical Inference.

2017-01-10T20:12:16-04:00

Math 502 Statistical Inference.
Spring 2015

Instructor: Xingye Qiao
Email: qiao@math.binghamton.edu
Phone number: (607) 777-2593
Office: WH 134
Meeting time & location: MWF 8:30 - 9:30 at WH 100E.
Office hours: MW 3:00 - 5:00

Prerequisite

Math 501.

Learning Objectives

Coverage: Chapters 6 through 10 in Casella & Berger.
Sufficiency, completeness, likelihood, estimation, testing, decision theory, Bayesian inference, sequential procedures, multivariate distributions and inference, nonparametric inference, asymptotic theory.

Recommended Texts

The required text is Casella & Berger (see below). Some reference texts are listed below as well.

Casella, G., & Berger, R. L. (2002). Statistical inference. Australia: Thomson Learning.
Lehmann, E. L. (1999). Elements of large-sample theory. New York: Springer.
Lehmann, E. L., & Casella, G. (1998). Theory of point estimation. New York: Springer.
Shao, J. (1999). Mathematical statistics. New York: Springer.
- Shao, J. (2005). Mathematical Statistics: Exercises and Solutions. New York, NY: Springer.
Hogg, R. V. and Craig, A. (1995). Introduction to Mathematical Statistics. Prentice Hall, Englewood Cliffs, NJ

Grading

Homework (40%): there will be weekly homework assignments, due at the beginning of each Wednesday class.
Midterm exams (20%+20%): there will be two midterm exams.
- Midterm exam 1: Friday, February 27, 2015
- Midterm exam 2: Friday, April 3, 2015
Final exam (20%): Wednesday, May 13, 2015 from 8:00 AM to 10:00 AM.

Homework assignments

Week 1

01/28: Notes, Example (2), (a)-(e); Textbook, Exercises 6.1, 6.3.
01/30: Notes, Example (4), b and c. Use both methods and for each method try to use different representations (so that your answers are not unique). Textbook, Exercises 6.2, 6.5, 6.8, 6.9.

Due on 02/04

Week 2

02/04 & 06: Textbook, Exercises 6.10, 6.11, 6.13, 6.14, 6.15, 6.18, 6.19, 6.20, 6.23, 6.25.
02/09:
1. Let $X_{1}\cdots X_{n}$ be i.i.d with density function defined as $f(x|\theta)=e^{-\lambda(x-\mu)},~x>\mu,~\lambda>0,~\mu\in \mathbb{R}$. Prove that $(X_{(1)},W)$ is the sufficient statistic of $\theta=(\mu,\lambda)$, where $W=\sum^{n}_{i=2}(X_{(i)}-X_{(1)})$.
2. Textbook: Exercises 7.1, 7.2, 7.6.

Due on 02/11

Week 3

02/11: Textbook, Exercises 7.7, 7.8, 7.10, 7.11.
02/13:
1. Notes: Show that a Bayes estimator depends on the data through a sufficient statistic.
2. Notes: if $X_i$'s are iid given $\theta$, are they iid marginally? Why?
3. Textbook, Exercises 7.14, 7.22, 7.23 (read “conjugate prior” as “prior”), 7.24, 7.25. Solutions to some of the questions may appear clearer after you read the lecture notes for Monday's class.
2/16: Textbook: Exercises 7.9, 7.12, 7.50.

Due on 02/18

Week 4

02/18: Textbook, Exercises 7.19, 7.20, 7.21.
02/20: Textbook, Exercises 7.37, 7.46, 7.49, 7.51, 7.52.
02/23: Textbook, Exercises 7.53, 7.57, 7.58.

Due on 02/25

Week 5

02/27: Textbook, Exercises 7.59, 7.44, 7.48, 7.60, 7.63.
03/02: Textbook, Exercises 7.40, 7.65, 7.66.

Due on 03/06

Week 6

03/06: Textbook, Exercises 8.1, 8.2, 8.3, 8.5, 8.6.
03/09: Textbook, Exercises 8.7, 8.8, 8.9.

Due on 03/11

Week 7

03/11: Textbook, Exercises 8.12, 8.13, 8.15, 8.17, 8.20.
03/13: Textbook, Exercises 8.19, 8.21.
03/16:
- Textbook, Exercises 8.25, 8.27.
- In the class, I showed that if we remove $k>0$ from the necessity condition of the NPL, then when $k=0$, we must have $\beta_\phi(\theta_1)=1$ for the UMP level $\alpha$ test $\phi$. Complete my work by arguing why in this case $\phi$ still must satisfy equation (2) in the NPL, that is, why it must be the case that $\phi=1$ when $f(x|\theta_1)>0$ wp1. [Hint: $\phi\le 1$. This proof should not last more than 2 lines.]

Due on 03/18

Week 8

03/18: Textbook, Exercises 8.28, 8.29, 8.30, 8.33.
03/20: Textbook, Exercises 8.37, 8.38, 8.47.
03/23: Textbook, Exercises 9.1, 9.2, 9.3.
Additional: Carry out the following simulation project. Submit the R code and report the result properly.
1. Use R to generate 10 observations from $N(1,4)$.
2. Now pretend that you only known that the data were from $N(\mu,4)$ without knowing $\mu$ and construct a 80% confidence interval for $\mu$.
3. Repeat Steps 1 and 2 100 times.
- Count the proportion among the 100 trials where the C.I. contains the true mean?
- What is the relation between the proportion and the confidence coefficient?
- Repeat Steps 1, 2 and 3, but pretend that you know neither the mean $\mu$ nor the variance $\sigma^2$. Then compare the lengths of the confidence intervals between the current and the previous settings. Make comments on the lengths and discuss why there is a difference.

Due on 03/25

Week 9

03/25: Submit all your code and output, preferably using LaTex. In a numerical problem, unless stated otherwise, $1-\alpha=0.95$. Textbook, Exercises
1. 9.4; In addition, assume that $n=10$ and $m=15$ and that $\sigma_X^2=1$ and $\sigma_Y^2=3$, generate some $X_i$'s and $Y_j$'s. Then use a numerical method to provide a CI based on the generated (observed) data. Then repeat the whole process for 1000 times. Report the number of time that the true $\lambda=3$ is covered by the CIs.
2. 9.6; here assume that $X\sim bin(n,p)$ is observed and $n$ is known. Next, let $n=50$ and generate $X$ with $p=0.3$. Numerically provide the CI for the observed $X$. Repeat for 1000 times and report the number of times that the true $p=0.3$ is covered by the CI.
3. 9.12.
4. 9.13(b).
03/27: Textbook, Exercises: 9.16, 9.17 and 9.23. In 9.17, you need to find the shortest confidence interval using the pivotal method (and prove it using a result in class). Moreover, find in addition a second CI using pivotal method with equal left and right probabilities. Assume that $\alpha=0.05$ and verify that your shortest confidence interval is indeed shorter than the second one.
03/30:
1. Textbook, Exercise: 9.37
2. Assume that $X_1,\dots,X_n$ are iid from Cauchy, where $f(x)=[\pi (1+x^2)]^{-1}$.
  1. Calculate $\int_{-\infty}^\infty |x|f(x)dx$.
  2. What is the mean of $X_1$?
  3. Can we apply the SLLN to prove that $\overline X_n\rightarrow \mu_X$ a.s.?
  4. Let $n=100$, simulate the sample and calculate $\overline X_n$. Then repeat this for 500 times. Collect all the $\overline X_n$'s and sort them (from the smallest to the greatest) and plot the sorted $\overline X_n$ values.

Due on 04/1

Week 10/11/12

03/27: Textbook, Exercises: 10.1, 10.2
Spring break
04/17:
1. Let $W_n$ be a random variable with mean $\mu$ and variance $C/n^\nu$ with $\nu>0$. Prove that $W_n$ is consistent with $\mu$.
2. Let $Y_n$ be the $n$th order statistic of a random sample of size $n$ from uniform$(0,\theta)$. Prove that $\sqrt{Y_n}$ is consistent with $\sqrt{\theta}$. Can you use Theorem 1 on page 51 of the lecture notes?
3. Let $Y_n$ be the $n$th order statistic of a random sample of size $n$ with continuous CDF $F(\cdot)$. Define $Z_n=n[1-F(Y_n)]$. Find the limiting distribution of $Z_n$. That is, is $Z_n$ convergent to some random variable, in what mode?
4. In the question above, let $F$ be the CDF for standard normal. Let $n$ be a large number. Then numerically verify you claim of the limiting distribution above by comparing $P(Z_n\le t)$ with $P(Z\le t)$ for arbitrary $t$ where $Z$ is the limiting random variable of $Z_n$.
5. In general, $X_n\Rightarrow X$ and $Y_n\Rightarrow Y$ cannot imply $X_n+Y_n\Rightarrow X+Y$. Please give a counterexample to illustrate this. The symbol $\Rightarrow$ means convergence in distribution.
04/20: Textbook,
1. Exercises: 10.4, 10.5, 10.6.
2. For $X\sim bin(n,p)$, let $\tau(p)=1/(1-p)$. What can we say about $\hat{\tau}$ for $p\ne 1$?
3. For $X_1,\dots,X_n\sim Unif(0,\theta)$, find the MLE of $\theta$. Find an unbiased estimator based which is a function of the MLE. Calculate the variance of this unbiased estimator. Calculate the theoretical optimal variance due to the CRLB. Compare them.

Due on 04/22

Week 13

4/22: Textbook. Exercises: 10.8, 10.19(a), 10.35.
4/24: Textbook. Exercises: 10.31, 10.32, 10.33, 10.34, 10.36, 10.37
4/27: In exercise 10.36, you were asked to derive two Wald statistics to run approximate large sample test. Now let $n=25$, $\alpha=1$, $H_0:\beta=\beta_0=2$. Please numerically compare the power of these two test when the true value of $\beta$ is 3, by running the test on the data for 10,000 times, and see which one rejects the null hypothesis more often. Try to interpret the result.

I am not satisfied with some of your answers to 9.23 in the homework returned today. I am giving a second chance for those who lost points for 9.23. You may submit your new answers (especially the numerical answers) along with this homework. I will consider adding back some points to that homework assignment. Please indicate that how many points you lost for 9.23. For the numerical answer, I have provided a Monte Carlo method to calculate the p value in the solution. You should use some other approach. For example, you can calculate the p value by taking the sum of the probabilities of $x$ which satisfies $LR(x)<LR(x_0)$ for $x=0,1,2,\dots,10000$ (instead of $\infty$) to approximate the p value, where $x_0$ is the observed data. This is just one suggestion and there are other approaches.

Due on 05/01

Week 14

4/29:
- Textbook. Exercises: 10.38.
- Suppose that a random variable $X$ has a Poisson distribution for which the mean $\theta$ is unknown. Find the Fisher information $I(\theta)$ in $X$.
- Suppose $X_1,\dots,X_n\sim Pois(\theta)$. Find the large sample $Z$ test, score test and LRT for testing $H_0:\theta=2$ vs $H_a:\theta\neq 2$.
- Simulate the distribution of $-2\log(\lambda_n)$ using the empirical distribution function (EDF) and compare it with the CDF of $\chi^2(1)$ distribution. You may revise the following code shown in the class to draw the EDF and CDF. Simulate a large number of data samples (say 5000), where each sample has size $n$. Make the case for $n=5$ and $n=100$.
5/1:
- Read Example 10.4.5 and finish exercise 10.40; finish exercise 10.41, 10.47 and 10.48.
- As in Example 10.3.4, with $\mathbf{X}\sim \textrm{Multinomial}(n,p_1,\ldots,p_5)$. Compare $H_0: p_1=p_2=p_5=0.01, p_3=0.5$ v.s. $H_1$: $H_0$ is not true.
  1. Derive the likelihood ratio test for $n=1$ and $n=100$ with level $\alpha=0.05$.
  2. Give an estimate of $P(H_o|H_1)$ when $p_1=p_2=p_5$, $p_3=0.3$, $n=100$, using simulation. Note that this is the probability of making type II error. Present the program.
  3. Compute $P(H_o|H_1)$ when $p_1=p_2=p_5$, $p_3=0.3$, $n=1$.
  4. Remark 1: in computing, sometimes it is better to use $\log(0^0)$ instead of $0*\log(0)$ as the latter can cause numerical trouble.
  5. Remark 2: What is the difference in degrees of freedom? Think how many additional constraints are imposed.
  6. Remark 3: You can try several combinations of $p_k$'s that satisfy $p_1=p_2=p_5$, $p_3=0.3$.

R code notes pp. 60, fig10.r

myfun=function(n){
	m=1000
	x=rgamma(m,n,1)/n # m X’s
	y=-2*(n*log(x)+n*(1-x)) # m λ’s
	u=rchisq(m,1)
	
	qqplot(y,u,main=paste("QQ plot, n=",n))
	lines(y,y)
	
	sy=sort(y)
	plot(sy,ppoints(sy), xlim=c(0.5,2), ylim=c(0.4,0.9), type="l", lty=1, main=paste("CDF, n=",n))
	lines(sy,pchisq(sy,1), xlim=c(0.5,2), ylim=c(0.4,0.9), type="l", lty=2)
}

pdf("fig10.pdf",height=9.0, width=6.5)
par(mfrow=c(2,2))
n=1
myfun(n)
n=100
myfun(n)
dev.off()

Due on 05/06

(Archive) Math 570 Applied Multivariate Analysis.

2016-02-28T17:51:35-04:00

Math 570 Applied Multivariate Analysis.
Fall 2014

Instructor: Xingye Qiao
Email: qiao@math.binghamton.edu
Phone number: (607) 777-2593
Office: OW-134
Meeting time & location: TR 10:05 - 11:30 at OW 100E.
Office hours: T 3:00-5:00, F 11:00-12:00
If you need to reach me, please e-mail qiao@math.binghamton.edu.
Please include [Math570] in the subject line of your email, or your email may not be read promptly.

Prerequisite

Math 501 and Math 502, or equivalent. Graduate students from outside of the mathematical department and senior undergraduate students may take this course with some preparation (please consult the instructor prior to the semester). One lecture session will be devoted to reviewing linear algebra materials that are useful in this course.

Learning Objectives

A review of the theoretical aspect of Multivariate Statistical Analysis, including: multivariate normal distributions, the multivariate Central Limit Theorem, quadratic forms, Wishart distributions, Hotelling's T square, inference about multivariate normal distributions.
Modern applied multivariate statistical methods, including: Principal Component Analysis, Canonical Correlation Analysis, Classification (Bayes rule, Linear and Quadratic discriminant analysis, cross-validation, and logistic regression etc.), factor analysis and Independent Component Analysis, clustering and multidimensional scaling.
Machine learning approaches, including Classification and Regression Trees, Support Vector Machine and other large margin classifiers, kernel methods, LASSO and sparsity methods, additive models, etc., if time permits.

Recommended Texts

The required texts are Härdle & Simar 2012 and Izenman 2013 (see below for details).

Elementary
- Johnson, Richard A & Wichern, Dean W. 2007. Applied multivariate statistical analysis. Upper Saddle River, N.J: Pearson Prentice Hall. Amazon Link
- Härdle, Wolfgang & Simar, Léopold. 2012. Applied multivariate statistical analysis. Berlin: Springer (also visit here for sample codes). Amazon Link

Advanced and applied
- Izenman, Alan Julian. 2013. Modern multivariate statistical techniques: Regression, classification, and manifold learning. New York: Springer New York. Amazon Link || Book Home Page (including R, S-plus and MATLAB code and data sets)
- Hastie, Trevor, Tibshirani, Robert, and Friedman, J. H. 2009. The elements of statistical learning: Data mining, inference, and prediction. New York, NY: Springer New York. Amazon Link
- James, Witten, Hastie and Tibshirani, 2014. An Introduction to Statistical Learning with Applications in R. Book Home Page. The PDF file of the book can be downloaded for free. There is also a R library for this book.

Theoretical
- Anderson, T. W. 2003. An introduction to multivariate statistical analysis. Hoboken, N.J: Wiley-Interscience. Amazon Link
- Muirhead, Robb J. 1982. Aspects of multivariate statistical theory. New York: Wiley. Amazon Link

Working with R or SAS
- Everitt, Brian, and Hothorn, Torsten. 2011. An introduction to applied multivariate analysis with R. New York: Springer. Amazon Link
- Khattree, Ravindra, and Naik, Dayanand N. 1999. Applied multivariate statistics with SAS software. Cary, NC: SAS Institute. Amazon Link
- Khattree, Ravindra, and Naik, Dayanand N. 2000. Multivariate data reduction and discrimination with SAS software. Cary, NC: SAS Institute. Amazon Link
- A Little Book of R for Multivariate Analysis

Most of the books listed above have been left on course reserve. You can go to the Newcomb Reading Room to loan the books for up to 1 day a time.

Grading

Homework (50%): there will be about four to five homework assignments.
Midterm exam (25% or 15%): a midterm exam focusing on the theoretical part of the course will be administered in the middle of the semester. Students who are not in the Math PhD program will receive a slightly easier set of problems and a smaller weight for the midterm exam than Math PhD students will do.
Presentation (10%): each student will choose a research topic (either original research or research conducted by other researchers) related to this course and give a 30-minute presentation. The presentation of each student shall be judged by peer students and the instructor.
Course project (15% or 25%): a final project will be assigned to each student by the end of the semester. Students who are not in the Math PhD program will gain a greater weight for the course project than Math PhD students will do. The guidelines for the final project can be found here.

Software

There is no designated software for this course. You may use the software that makes the most sense for you. Many pharmaceutical companies use SAS for compliance with FDA regulations. Academic intuitions as well as labs often use R and python. Corporations often use MATLAB, Stata, Minitab, S, etc. because of the relatively high reliability despite the cost. However, it is expected that the student immerse herself with use of at least one software.

Used to be expensive, SAS University Edition is now free for download and use.

Course decks

You need to log in this website in order to download these documents. They can also be downloaded from the blackboard (see the content page).

Multivariate Data Exploration, slides, Reading: HS Ch. 1, Izenman Ch. 4
Matrix algebra review, notes, Reading: HS Ch. 2, Izenman Sec. 3.2
Multivariate normal distribution, notes, Reading: HS Ch. 4-5, Izenman Sec. 3.3
The Wishart distribution, notes, Reading: HS Ch. 4-5, Izenman Sec. 3.4
Inference for MVN, notes, Reading: HS Ch. 6-7, Izenman Sec. 3.5
Linear Dimension Reduction: PCA and CCA, slides, Reading: HS Ch. 9-10, 15, Izenman Ch 7
Classification, slides, Reading: HS Ch. 13, Izenman 8 & ESL Ch. 4.3, 4.4, and 7.10.
Linear Dimension Reduction: Latent Variable Models, slides, Reading: HS Ch. 11, Izenman 15
Clustering, slides, Reading: HS Ch. 12, Izenman 12
Multidimensional Scaling, slides, Reading: HS 16, Izenman 13
Support Vector Machine, slides, Reading: Izenman 11, ESL 12
Sparsity, slides, Reading: Izenman 5.6-5.8, ESL 3
Tree, slides, Reading: Izenman 9, ESL 9.2
Ensemble methods, slides, Reading: Izenman 14, ESL 7.11, 8.2, 10, 16.

Schedule

Week 1 (Sept. 2, 4): introduction; matrix
Week 2 (Sept. 9, 11): MVN; Wishart
Week 3 (Sept. 16, 18): Wishart; Inference for MVN; PCA
Week 4 (Sept. 23): PCA; (break)
Week 5 (Sept. 30, Oct. 2): CCA; Bayes rule, LDA/QDA
Week 6 (Oct. 7, 9): Logistic Regression; cross validation; other classifiers
Week 7 (Oct. 14, 16): FA; ICA
Week 8 (Oct. 21, 23): k-means; Hierarchical clustering
Week 9 (Oct. 28, 30): Gaussian mixture/EM; manova
Week 10 (Nov. 4, 6): ; mds; svm;
Week 11 (Nov. 11, 13): svm; kernel methods; lasso
Week 12 (Nov. 18, 20): lasso; midterm exam
Week 13 (Nov. 25, 27): tree; (break)
Week 14 (Dec. 2, 4): bootstrap, bagging, subsampling; boosting; RF
Week 15 (Dec. 9, 11): high dimensional learning; multiple testing
Week 16: final week

Presentations

Nov 4	Xiaojie Du (zou06)
Nov 6	Lin Yao (shen07)
Nov 11	Lishun Li (Jung09)
Nov 13	Armin Pillhofer (tibshirani02)
Nov 18	Zach Seymour (Witten10)
Nov 20	Ruiqi Liu (sun12)

(Archive) Math 570 Applied Multivariate Analysis.

2017-01-10T20:11:26-04:00

Math 570 Applied Multivariate Analysis.
Spring 2016

Instructor: Xingye Qiao
Email: qiao@math.binghamton.edu
Phone number: (607) 777-2593
Office: WH-134
Meeting time & location: MWF 10:50 – 11:50 at WH-100E.
Office hours: MW 3–4 and F 9:45 –10:45
If you need to reach me, please e-mail qiao@math.binghamton.edu.
Please include [Math570] in the subject line of your email, or your email may not be read promptly.

Prerequisite

Learning Objectives

A review of the theoretical aspect of Multivariate Statistical Analysis, including: multivariate normal distributions, the multivariate Central Limit Theorem, quadratic forms, Wishart distributions, Hotelling's T square, inference about multivariate normal distributions.
Modern applied multivariate statistical methods, including: Principal Component Analysis, Canonical Correlation Analysis, Classification (Bayes rule, Linear and Quadratic discriminant analysis, cross-validation, and logistic regression etc.), factor analysis and Independent Component Analysis, clustering and multidimensional scaling.
Machine learning approaches, including Classification and Regression Trees, Support Vector Machine and other large margin classifiers, kernel methods, LASSO and sparsity methods, additive models, etc., if time permits.

Recommended Texts

The required texts are Härdle & Simar 2012 and Izenman 2013 (see below for details).

Elementary
- Johnson, Richard A & Wichern, Dean W. 2007. Applied multivariate statistical analysis. Upper Saddle River, N.J: Pearson Prentice Hall. Amazon Link
- Härdle, Wolfgang & Simar, Léopold. 2012. Applied multivariate statistical analysis. Berlin: Springer (also visit here for sample codes). Amazon Link

Advanced and applied
- Izenman, Alan Julian. 2013. Modern multivariate statistical techniques: Regression, classification, and manifold learning. New York: Springer New York. Amazon Link || Book Home Page (including R, S-plus and MATLAB code and data sets)
- Hastie, Trevor, Tibshirani, Robert, and Friedman, J. H. 2009. The elements of statistical learning: Data mining, inference, and prediction. New York, NY: Springer New York. Amazon Link
- James, Witten, Hastie and Tibshirani, 2014. An Introduction to Statistical Learning with Applications in R. Book Home Page. The PDF file of the book can be downloaded for free. There is also a R library for this book.

Theoretical
- Anderson, T. W. 2003. An introduction to multivariate statistical analysis. Hoboken, N.J: Wiley-Interscience. Amazon Link
- Muirhead, Robb J. 1982. Aspects of multivariate statistical theory. New York: Wiley. Amazon Link

Working with R or SAS
- Everitt, Brian, and Hothorn, Torsten. 2011. An introduction to applied multivariate analysis with R. New York: Springer. Amazon Link
- Khattree, Ravindra, and Naik, Dayanand N. 1999. Applied multivariate statistics with SAS software. Cary, NC: SAS Institute. Amazon Link
- Khattree, Ravindra, and Naik, Dayanand N. 2000. Multivariate data reduction and discrimination with SAS software. Cary, NC: SAS Institute. Amazon Link
- A Little Book of R for Multivariate Analysis

Grading

Homework (50%): there will be about four to five homework assignments.
Midterm exam (20% or 15%): a midterm exam focusing on the theoretical part of the course will be administered in the middle of the semester. Students who are not in the Math PhD program will receive a slightly easier set of problems and a smaller weight for the midterm exam than Math PhD students will do.
Presentation (10%): each student will choose a research topic (either original research or research conducted by other researchers) related to this course and give a 30-minute presentation. The presentation of each student shall be judged by peer students and the instructor.
Course project (15% or 20%): a final project will be assigned to each student by the end of the semester. Students who are not in the Math PhD program will gain a greater weight for the course project than Math PhD students will do. The guidelines for the final project can be found here.
Lecture attendance and participation (5%)

Software

Used to be expensive, SAS University Edition is now free for download and use.

Schedule

The course deck may be downloaded from the blackboard.

Week 1 (Jan 25 - 29)
- Multivariate Data Exploration, Reading: HS Ch. 1, Izenman Ch. 4.
- Matrix algebra review, Reading: HS Ch. 2, Izenman Sec. 3.2.
Week 2 (Feb 1 - 5)
- Random vectors and multivariate normal distribution; The Wishart distribution
Week 3 (Feb 8 - 12)
- The Wishart distribution; Inference about multivariate normal distribution
Week 4 (Feb 15 - 19)
- PCA
Week 5 (Feb 22 - 26)
- CCA; Classification: Bayes rule
Week 6 (Feb 29 - Mar 4)
- LDA, QDA, Logistic regression
Week 7 (Mar 7 - Mar 11)
- Cross validation and other classifiers; FA;
Week 8 (Mar 14 - Mar 18)
- ICA; k-means; Hierarchical clustering
Week 9 (Mar 21 - Mar 25)
- Gaussian mixture/EM
Week 10 (Mar 28 - Apr 1)
- Spring Break
Week 11 (Apr 4 - Apr 8)
- MDS; SVM
Week 12 (Apr 11 - Apr 15)
- 4 presentations and midterm exam.
Week 13 (Apr 18 - Apr 22)
- LASSO
Week 14 (Apr 25 - Apr 29)
- Tree methods
Week 15 (May 2 - May 6)
- bootstrap, bagging, subsampling; boosting; RF
Week 16 (May 9 - May 11)
- final projects.

Student Presentations

Feb 26: Xu Chu presents Zou et al. 2006, “Sparse Principal Component Analysis.”
Feb 29: Tianqi Zhang presents Shen and Huang 2007, “Sparse principal component analysis via regularized low rank matrix approximation.”
Mar 4: Liping Gu presents Tibshirani et al. 2002, “Diagnosis of multiple cancer types by shrunken centroids of gene expression.”
Mar 7: Xin Gu presents Fan and Fan 2008, “High-dimensional classification using features annealed independence rules.”
Mar 11: Haomiao Meng presents Bickel and Levina, 2004, “Some theory for Fisher's linear discriminant function, `naive Bayes', and some alternatives when there are many more variables than observations.”
Mar 16: Baiyang Qi presents Ahn and Marron, 2008, “The maximal data piling direction for discrimination.”
Mar 21: Rui Gao presents Witten and Tibshirani, 2011, “Penalized classification using Fisher's linear discriminant.”
Mar 25: Yinsong Chen presents Mai and Zou 2013, “A Note On the Connection and Equivalence of Three Sparse Linear Discriminant Analysis Method.”
Apr 4: Hao Xu presents Zhu and Hastie, 2004, “Classification of gene microarrays by penalized logistic regression.”
Apr 11: Sreedhar Kumar presents Sun and Wang, 2012, “Regularized k-means clustering of high-dimensional data and its asymptotic consistency.”
Apr 11: Yuan Fang presents Witten and Tibshirani, 2010, “A framework for feature selection in clustering.”
Apr 13: Wenming Deng presents Liu et al., 2008, “Statistical Significance of Clustering for High-Dimension, Low-Sample Size Data.”
Apr 13: Liang Chen presents Jung and Qiao 2014, “A statistical approach to set classification by feature selection with applications to classification of histopathology images.”
Apr 22: Junle Lu presents Zhu and Hastie, 2005, “Kernel logistic regression and the import vector machine.”
Apr 29: Miaolin Fan presents Wang, Nan, Rosset and Zhu, 2011, “Random lasso.”
May 2: Xiang Li presents Qiao and Zhang 2015, “Distance-weighted Support Vector Machine.”

Teach

2026-06-30T05:11:35-04:00

Teach

Spring 2017
- Math 570 Applied Multivariate Analysis.
Spring 2016
- (Archive) Math 448 Mathematical Statistics.
- (Archive) Math 570 Applied Multivariate Analysis.
Fall 2015
- (Archive) Math 448 Mathematical Statistics.
Spring 2015
- (Archive) Math 502 Statistical Inference.
Fall 2014
- (Archive) Math 448 Introduction to Probability and Statistics II.
- (Archive) Math 570 Applied Multivariate Analysis.