Course Code: MATH 3465
Course Title: Statistical Inference
Course Type: Core
Level: 3
Semester: 2
No. of Credits: 3
Prerequisite(s): MATH 2275 (Statistics I) and (MATH 2270) Multivariable Calculus
Course Rationale
The course is aimed at those whose future careers will involve a heavy use of statistical methods and at the same time fulfil the required mathematical statistical inference needed at the undergraduate level. The course will benefit future statisticians, economists, geneticists and mathematicians with an interest in Statistics. The course has two main goals:
 To give students a firm and comprehensive knowledge of basic Frequentist and Bayesian Inference.
 To introduce main ideas of nonparametric inference based on the empirical distribution function.
The course is necessary to provide the mathematical statistical foundation for more advanced courses in programmes in statistics in the Department of Mathematics and Statistics. The exposure of students to this course gives them the ability to become good problem solvers. This course therefore creates the opportunity for our graduates to become critical and creative thinkers – thereby fulfilling one of the key attributes of our graduates.
Course Description
This is a second course in Statistical Theory. The course may be thought of as a direct continuation of the introductory second year course Statistics I. This course is necessary to expose students to both classical and Bayesian inference which they would not have encountered in Statistics I. While Statistics I gives a relatively broad nontheoretical approach to statistics, this course completes the undergraduate statistical theory so that students can understand the underlying concepts in a more concise mathematical setting.
The course consists of three fairly distinct modules–frequentist inference, Bayesian inference and nonparametric methods. We continue the discussion of classical inference begun in Math 2275 Likelihood techniques are applied to a wide range of models. There is a fairly detailed discussion of unbiasedness and sufficiency. UMP and likelihood ratio tests are discussed. For Bayesian Inference, we introduce the ideas of subjective probability, prior and posterior distributions and the basics of Bayesian estimation and testing. In the short section on nonparametric methods we introduce the empirical distribution function and tests based on it. There is a brief introduction to inference on censored data and an introduction to the bootstrap.
All lectures, assignments, handouts, and review materials are available online through myeLearning to all students. Blended leaning techniques will be employed. Lectures will be supplemented with laboratory work and group discussions.
Assessment is designed to encourage students to work continuously with the course materials. Active learning will be achieved through weekly assignments and problem sheets allowing continuous feedback and guidance on problem solving techniques in tutorials and lectures. Assessment will be based on the weekly assignments and incourse tests followed by a final examination based on the whole course.
Learning Outcomes
Upon successful completion of the course, students will be able to:
 Derive maximum likelihood estimates of parameters from the main distributions and in linear models and state the main properties of maximum likelihood estimators.
 Calculate the Fisher Information in an observation and determine whether an estimator is UMVUE.
 Determine whether a statistic is sufficient.
 Explain the importance of sufficient statistics.
 Derive best tests of a simple null versus a simple alternative.
 Define a UMP test and be able to show that a test is UMP.
 Derive the likelihood ratio tests of parameters in various linear models.
 Explain the concepts of prior and posterior distributions.
 Calculate the posterior distribution of a parameter.
 Find the Bayes estimate of a parameter.
 Test for equality of two distributions.
 Calculate the KaplanMeier Estimate of a survival function.
 Find bootstrap estimates of parameters in simple situations.
Content
 Method of moments estimators. Maximum likelihood estimators of parameters in various oneparameter and multiparameter distributions. MLEs of parameters in linear models. Desirable properties of MLEs.
 Unbiased estimators. Fisher Information and the CramerRao Inequality, including proofs.
 Sufficiency and joint sufficiency. The Fisher Factorization Criterion. The RaoBlackwell theorem with proof.
 The NeymanPearson theory of hypothesis testing. The NeymanPearson theorem with proof. Uniformly most powerful tests. Tests with monotone likelihood ratio.
 Likelihood Ratio Tests. Derivation of usual Ftests in linear models as likelihood ratio tests.
 Subjective probability. Prior and posterior distributions of parameters. Conjugate priors. Bayes estimators. Bayes tests.
 The Empirical Distribution Function.
 KolmogorovSmirnov Tests.
 Censored data and the KaplanMeier Estimator.
Teaching Methodology
Lectures, tutorials, assignments and problem papers.
Lectures: Two lectures per week.
Labs: One two hour computer lab per week.
Assignments: One assignment (marked) per week.
Additional problems will be given during lectures and tutorials but will not be marked. However, students will need to do some of these, as well as the assignments, in order to learn the material properly and to adequately prepare for examinations and quizzes.
Assessment
Coursework Mark: 50%, based on three interm examinations of 12% each and assignments/lab work weighted at 14%.
Final Examination: 50% (one 2hour written paper).
Course Calendar
WEEK 
LECTURE TOPIC 
ASSIGNMENTS 
TESTS 

1 
Introduction/Course Overview Method of moments estimators. Maximum likelihood estimation. Maximum Likelihood estimators of parameters in linear models. 
Assignment 1 

2 
Maximum Likelihood estimators of parameters in linear models. Desirable properties of mles 
Assignment 2 Assignment 1 returned. 

3 
. Sufficiency and joint sufficiency. Fisher factorization criterion. The RaoBlackwell theorem. Exponential families. 
Assignment 3 Assignment 2 returned. 

4 
Unbiased estimators. Fisher Information and CramerRao inequality with proof 
Assignment 4 Assignment 3 returned 

5 
The NeymanPearson lemma. Statement and examples. Proof of the NeymanPearson Lemma. 
Assignment 5 Assignment 4 returned 

6 
Uniformly most powerful tests. Tests with monotone likelihood ratios. Likelihood Ratio Tests. 
Assignment 6 Assignment 5 returned 
Test 1 on material in weeks 14. 
7 
LR tests of parameters in linear models 
Assignment 7 Assignment 6 Returned 

8 
Prior and posterior distributions. Conjugate priors. Bayes estimators. 
Assignment 8 Assignment 7 returned 

9 
Testing in the Bayesian Framework. 
Assignment 9 Assignment 8 returned 
Test 2 on material from weeks 57 
10 
The Empirical Distribution Function. Onesample and twosample KolmogorovSmirnov tests. 
Assignment 10 Assignment 9 returned 

11 
The Bootstrap. 
Assignment 11 Assignment 10 returned 

12 
The Bootstrap Continued. 
Assignment 12 Assignment 11 returned 
Test 3 on material from 89 
13 
Revision 
Revision Assignment 12 returned 

Reference Material
Prescribed Text
Statistical Inference by Casella and Berger, 2^{nd} edition, Wiley.
Other Recommended Texts
Mathematical Statistics by Hogg and Craig (Wiley).
Probability and Statistics 3^{rd} edition by Morris de Groot and Schervish
(AddisonWesley 2002)
Software
Some of the assigned exercises will require the use of R, Minitab and SPSS statistical packages. R is free statistical software. Students can download R and use it on their personal computers. Minitab is free for UWI students. SPSS can be accessed at the computer labs.