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Math 786: Large Sample Theory and Inference
Fall 2017 Graduate Course Syllabus

NJIT Academic Integrity Code: All Students should be aware that the Department of Mathematical Sciences takes the University Code on Academic Integrity at NJIT very seriously and enforces it strictly. This means that there must not be any forms of plagiarism, i.e., copying of homework, class projects, or lab assignments, or any form of cheating in quizzes and exams. Under the University Code on Academic Integrity, students are obligated to report any such activities to the Instructor.

Course Information

Course Description: Limit theorems, central limit theorem, asymptotic expansions and large deviations, limit theorems in martingales and semi-martingales and stochastic differential equations, asymptotic expansions of functions of statistics, linear parametric estimation, asymptotic efficiency, martingale approach to inference: test for homogeneity and goodness of fit, decomposable statistics, inference for counting processes and censored data, inference in nonlinear regression, existence and consistency of least squares estimator (LSE), asymptotic properties of LSE, Von Mises functionals, estimation of parameters of stable laws, empirical characteristics function for inference, generalized least squares for linear models.

Number of Credits: 3

Prerequisites: MATH 662.

Course-Section and Instructors

Course-Section Instructor
Math 786-001 Professor S. Subramanian

Office Hours for All Math Instructors: Fall 2017 Office Hours and Emails

Required Textbooks:

Title Approximation Theorems of Mathematical Statistics
Author Serfling
Edition 1st
Publisher Wiley
ISBN # 978-0471219279

University-wide Withdrawal Date:The last day to withdraw with a w is Monday, November 6, 2017. It will be strictly enforced.

Course Goals/ EXTRA INFORMATION

Course Objectives and Description: This course will be concerned with teaching doctoral level mathematical sciences students the rudiments of statistical large sample theory, focusing on a variety of limit theorems and techniques useful in mathematical statistics. Topics covered will be asymptotic distributions of estimators such as sample moments and quantiles, order statistics, and maximum likelihood; projection and U statistics; empirical processes; functional delta method; and nonparametric density estimation.

Course Outcomes

Course Assessment: Will be based on regular homework, one midterm exam, and a final in-class presentation of a course-specific rigorous research topic assigned by instructor.

Policies

DMS Course Policies: All DMS students must familiarize themselves with, and adhere to, the Department of Mathematical Sciences Course Policies, in addition to official university-wide policies. DMS takes these policies very seriously and enforces them strictly.

Grading Policy: The final grade in this course will be determined as follows:

Homework 30%
Midterm Exam 40%
Final Exam 30%

Your final letter grade will be based on the following tentative curve.

A 85 - 100 C+ 70 - 74
B+ 80 - 84 C 60 - 73
B 75 - 79 F 0 - 59

Attendance Policy: Attendance at all classes will be recorded and is mandatory. Please make sure you read and fully understand the Math Department’s Attendance Policy. This policy will be strictly enforced.

Homework Policy: Homework assignments are due within a week unless announced otherwise by instructor.  Late homework will not be accepted.

Exams: There will be one midterm exam held in class during the semester and one comprehensive final exam. Exams are held on the following days:

Midterm Exam November 8, 2017
Final Exam Period December 15 - 21, 2017

The final exam will test your knowledge of all the course material taught in the entire course. Make sure you read and fully understand the Math Department's Examination Policy. This policy will be strictly enforced.

Makeup Exam Policy: To properly report your absence from a midterm or final exam, please review and follow the required steps under the DMS Examination Policy found here:

Cellular Phones: All cellular phones and other electronic devices must be switched off during all class times.

Additional Resources

Accommodation of Disabilities: Disability Support Services (DSS) offers long term and temporary accommodations for undergraduate, graduate and visiting students at NJIT.

If you are in need of accommodations due to a disability please contact Chantonette Lyles, Associate Director of Disability Support Services at 973-596-5417 or via email at lyles@njit.edu. The office is located in Fenster Hall, Room 260.  A Letter of Accommodation Eligibility from the Disability Support Services office authorizing your accommodations will be required.

For further information regarding self identification, the submission of medical documentation and additional support services provided please visit the Disability Support Services (DSS) website at:

Important Dates (See: Fall 2017 Academic Calendar, Registrar)

Date Day Event
September 5, 2017 T First Day of Classes
September 11, 2017 M Last Day to Add/Drop Classes
November 6, 2017 M Last Day to Withdraw
November 21, 2017 T Thursday Classes Meet
November 22, 2017 W Friday Classes Meet
November 23 - 26, 2017 R - Su Thanksgiving Break - University Closed
December 13, 2017 W Last Day of Classes
December 14, 2017 R Reading Day
December 15 - 21, 2017 F - R Final Exam Period

Course Outline

Week Lecture Topic
1 Week of 9/4 PRELIMINARY TOOLS AND FOUNDATIONS
Inverse cumulative distribution function, characteristic functions, modes of convergence, uniformly-integrable sequence, convergence of transformed sequences
2 Week of 9/11 PRELIMINARY TOOLS AND FOUNDATIONS -- CONTINUED 
Laws of large numbers, central limit theorem, law of the iterated logarithm, Berry-Esseen theorem 
3 Week of 9/18 PRELIMINARY TOOLS AND FOUNDATIONS -- CONTINUED 
Empirical distribution functions, asymptotic normality, delta method
4 Week of 9/25 SAMPLE MOMENTS
Joint asymptotic distribution of sample moments
5 Week of 10/2 SAMPLE QUANTILES - I
Law of large numbers and central limit theorem for sample quantiles
6 Week of 10/9 SAMPLE QUANTILES - II 
Joint asymptotic distribution of sample quantiles, Bahadur representation
7 Week of 10/16 OTHER ASYMPTOTICS - I
Asymptotic multivariate normality of cell frequency vectors, sample quantile process
8  Week of 10/23 OTHER ASYMPTOTICS - II
Limit distribution of the chi-squared goodness-of-fit statistic, consistency  of maximum likelihood estimates
9 Week of 10/30 PROJECTIONS AND U-STATISTICS
Projections, conditional expectation, projection onto sums, Hoeffding decomposition
10 Week of 11/6 MIDTERM EXAM
Material taught up to the end of week 8 will be tested in the exam
11 Week of 11/13 PROJECTIONS AND U-STATISTICS -- CONTINUED
One-sample, two-sample, and degenerate U-statistics
12 Week of 11/20 EMPIRICAL PROCESSES - I 
Empirical distribution functions, empirical distributions
13 Week of 11/27 EMPIRICAL PROCESSES – II
Goodness-of-fit statistics, random functions
14 Week of 12/4 (Thurs) EMPIRICAL PROCESSES - III
Changing classes, maximal inequalities
15 Week of 12/11 STUDENT PRESENTATIONS

Updated by Professor S. Subramanian - 9/1/2017
Department of Mathematical Sciences Course Syllabus, Fall 2017