## STATS 3006 - Mathematical Statistics III

### North Terrace Campus - Semester 1 - 2023

Statistical methods used in practice are based on a foundation of statistical theory. One branch of this theory uses the tools of probability to establish important distributional results that are used throughout statistics. Another major branch of statistical theory is statistical inference. It deals with issues such as how do we define a "good" estimator or hypothesis test, how do we recognise one and how do we construct one? This course is concerned with the fundamental theory of random variables and statistical inference. Topics covered are: calculus of distributions, moments, moment generating functions; multivariate distributions, marginal and conditional distributions, conditional expectation and variance operators, change of variable, multivariate normal distribution, exact distributions arising in statistics; weak convergence, convergence in distribution, weak law of large numbers, central limit theorem; statistical inference, likelihood, score and information; estimation, minimum variance unbiased estimation, the Cramer-Rao lower bound, exponential families, sufficient statistics, the Rao-Blackwell theorem, efficiency, consistency, maximum likelihood estimators, large sample properties; tests of hypotheses, most powerful tests, the Neyman-Pearson lemma, likelihood ratio, score and Wald tests, large sample properties.

• General Course Information
##### Course Details
Course Code STATS 3006 Mathematical Statistics III Mathematical Sciences Semester 1 Undergraduate North Terrace Campus 3 Up to 3 hours per week Y (MATHS 1012 and STATS 2107) or (MATHS 2201 and MATHS 2202) or (MATHS 2106 and MATHS 2107) STATS 2107 Statistical methods used in practice are based on a foundation of statistical theory. One branch of this theory uses the tools of probability to establish important distributional results that are used throughout statistics. Another major branch of statistical theory is statistical inference. It deals with issues such as how do we define a "good" estimator or hypothesis test, how do we recognise one and how do we construct one? This course is concerned with the fundamental theory of random variables and statistical inference. Topics covered are: calculus of distributions, moments, moment generating functions; multivariate distributions, marginal and conditional distributions, conditional expectation and variance operators, change of variable, multivariate normal distribution, exact distributions arising in statistics; weak convergence, convergence in distribution, weak law of large numbers, central limit theorem; statistical inference, likelihood, score and information; estimation, minimum variance unbiased estimation, the Cramer-Rao lower bound, exponential families, sufficient statistics, the Rao-Blackwell theorem, efficiency, consistency, maximum likelihood estimators, large sample properties; tests of hypotheses, most powerful tests, the Neyman-Pearson lemma, likelihood ratio, score and Wald tests, large sample properties.
##### Course Staff

Course Coordinator: Dr Melissa Humphries

##### Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
##### Course Learning Outcomes
On successful completion of this course students will be able to:
1. demonstrate knowledge of, and properties of, statistical models in common use,
2. understand the basic principles underlying statistical inference (estimation and hypothesis testing),
3. be able to construct tests and estimators, and derive their properties,
4. demonstrate knowledge of applicable large sample theory of estimators and tests.

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)

Attribute 1: Deep discipline knowledge and intellectual breadth

Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.

1,2,3,4

Attribute 2: Creative and critical thinking, and problem solving

Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.

1,2,3

Graduates engage in professional behaviour and have the potential to be entrepreneurial and take leadership roles in their chosen occupations or careers and communities.

1,2,3

Attribute 8: Self-awareness and emotional intelligence

Graduates are self-aware and reflective; they are flexible and resilient and have the capacity to accept and give constructive feedback; they act with integrity and take responsibility for their actions.

1,2
• Learning Resources
##### Required Resources
Lecture slides will be provided on MyUni
##### Recommended Resources
Recommended textbooks:

Mathematical Statistics with Applications (7th ed.), by D.D. Wackerly, W. Mendenhall, and R.L. Scheaffer, Duxbury Press.
Mathematical Statistics and Data Analysis (3rd ed.), by J.A. Rice, Duxbury Press.

Useful textbooks:

Statistical Inference (2nd ed.), by G. Casella and R. L. Berger, Duxbury Press.
Modern Mathematical Statistics with Applications (2nd ed.), by J.L. Devore and K.N. Berk, Springer.

##### Online Learning
This course uses MyUni exclusively for providing electronic resources: lecture notes, assignments, solutions, etc. Students are advised to make extensive use of these resources.
• Learning & Teaching Activities
##### Learning & Teaching Modes
This course relies on workshops as the primary delivery mechanism for the material. Tutorials supplement the workshops by providing exercises and example problems to enhance the understanding obtained through lectures. A sequence of written assignments provides assessment opportunities for students to gauge their progress and understanding.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

 Activity Quantity Workload Hours Workshops 24 75 Tutorials 11 33 Assessments 14 48 Total 156
##### Learning Activities Summary
Workshops will cover:

Transfromations of random variables and their moments
Marginal and conditional distributions and their moments
Joint distributions, covariance, correlation, independence of random variables, distributions of functions of jointly distributed random variables, conditional distributions, conditional means and variances
Sums of independent random variables, transformations of two or more jointly distributed random variables
Random vectors, the multivariate normal distribution and properties
Modes of convergence, laws of large numbers, central limit theorem, Jensen's inequality
Random samples, the chi-square, t, and F distributions and their roles in normal sampling, basic concepts of statistical inference, the likelihood principle, sufficient statistics
Basic concepts of estimation; method of moments, maximum likelhood, large sample properties (consistency, asymptotic normality), mean square eror, Rao-Blackwell theorem
Fisher information, the Cramer-Rao inequality, confidence intervals and properties
Hypothesis testing, types of errors, p-value, power, Neyman-Pearson lemma, uniformly most powerful tests, likelihood ratio tests, Wald tests, score tests

• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

##### Assessment Summary
 Component Weighting Objective Assessment Assignments 30% all Quizzes 10% all Mid-Semester Test 20% all Exam 40% all
##### Assessment Related Requirements
An aggregate score of at least 50% is required to pass the course.
##### Assessment Detail
There are three assignments in this course (each contribute 10% of final grade).
There are also 10 quizzes througout the course (1% each).
There will be a mid-semester test (20%) and a final exam (40% of final grade).
##### Submission
1. All written assignments are to be submitted online via MyUni.
2. Late assignments will not be accepted unless an extension has been arranged prior to the due date.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
FNS   Fail No Submission
F 1-49 Fail
P 50-64 Pass
C 65-74 Credit
D 75-84 Distinction
HD 85-100 High Distinction
CN   Continuing
NFE   No Formal Examination
RP   Result Pending

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
• Fraud Awareness

Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's studentâ€™s disciplinary procedures.

The University of Adelaide is committed to regular reviews of the courses and programs it offers to students. The University of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.

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