STATS 3006 - Mathematical Statistics III

North Terrace Campus - Semester 1 - 2019

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
    Course Mathematical Statistics III
    Coordinating Unit School of Mathematical Sciences
    Term Semester 1
    Level Undergraduate
    Location/s North Terrace Campus
    Units 3
    Contact Up to 3 hours per week
    Available for Study Abroad and Exchange Y
    Prerequisites STATS 2107 or (MATHS 2201 and MATHS 2202)
    Assumed Knowledge STATS 2107
    Course Description 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

    No information currently available.

    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.

    University Graduate Attributes

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

    University Graduate Attribute Course Learning Outcome(s)
    Deep discipline knowledge
    • informed and infused by cutting edge research, scaffolded throughout their program of studies
    • acquired from personal interaction with research active educators, from year 1
    • accredited or validated against national or international standards (for relevant programs)
    1,2,3,4
    Critical thinking and problem solving
    • steeped in research methods and rigor
    • based on empirical evidence and the scientific approach to knowledge development
    • demonstrated through appropriate and relevant assessment
    1,2,3
    Career and leadership readiness
    • technology savvy
    • professional and, where relevant, fully accredited
    • forward thinking and well informed
    • tested and validated by work based experiences
    1,2,3
    Self-awareness and emotional intelligence
    • a capacity for self-reflection and a willingness to engage in self-appraisal
    • open to objective and constructive feedback from supervisors and peers
    • able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate
    1,2
  • Learning Resources
    Required Resources
    A set of lecture notes will be provided.
    Recommended Resources
    Online (MyUni) resources:
    1. Mathematical Statistics III Lecture Notes, by P. Solomon and G Glonek.
    2. Introduction to Statistical Theory, by R. Muirhead and J. Sun.

    Useful books:
    1. Mathematical Statistics and Data Analysis (3rd ed.), by J. A. Rice, Duxbury Press.
    2. Statistical Inference (2nd ed.), by G. Casella and R. L. Berger, Duxbury Press.
    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 lectures as the primary delivery mechanism for the material. Tutorials supplement the lectures 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.
    Workload

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


    Activity Quantity Workload Hours
    Lectures 30 90
    Tutorials 5 18
    Assignments 5 48
    Total 156
    Learning Activities Summary
    Lecture outline

    1-3: Review of probability, random variables, density and mass functions, expectation, mean, variance
    4-6: Standard probability distributions (statistical models) and their properties
    6-7: Exponential families of distributions; distribution and expectation of a function of a random variable
    8-11: Joint distributions, covariance, correlation, independence of random variables, distributions of functions of jointly distributed random variables, conditional distributions, conditional means and variances
    12-14: Sums of independent random variables, transformations of two or more jointly distributed random variables
    14-15: Random vectors, the multivariate normal distribution and properties
    16-19: Modes of convergence, laws of large numbers, central limit theorem, Jensen's inequality
    20-22: 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
    23-25: Basic concepts of estimation; method of moments, maximum likelhood, large sample properties (consistency, asymptotic normality), mean square eror, Rao-Blackwell theorem
    26-27: Fisher information, the Cramer-Rao inequality, confidence intervals and properties
    28-30: Hypothesis testing, types of errors, p-value, power, Neyman-Pearson lemma, uniformly most powerful tests, likelihood ratio tests, Wald tests, score tests

    Tutorial outline: Tutorial material will be integrated into the lecture and assignment material


  • 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
    Exam 70% all
    Assessment Related Requirements
    An aggregate score of at least 50% is required to pass the course.
    Assessment Detail
    Five equally weighted (6% each) assigments, due on Friday by 4 pm at the end of weeks 3, 5, 7, 9, 12.  The assignments will be distributed on Monday of weeks 2, 4, 6, 8, 10.
    Submission
    1. All written assignments are to be submitted to the designated hand-in box in the School of Mathematical Sciences with a signed cover sheet attached.
    2. Late assignments will not be accepted.
    3. Assignments will have a two-week turn-around time for feedback to students.
    Course Grading

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

    M10 (Coursework Mark Scheme)
    Grade Mark Description
    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
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