Global Opportunities

MAS2904 : Statistical Inference & Stochastic Modelling

Semesters
Semester 1 Credit Value: 10
Semester 2 Credit Value: 10
ECTS Credits: 10.0

Aims

To lay the foundations of statistical inference. The students will learn about the distinction between a population and a sample. They will know about the use of estimators calculated from random samples as a means of learning about properties of the population. They will be able to describe the role of likelihood methods in the derivation of estimators and their properties.

This module will also provide an introduction to ways of mathematically describing processes that exhibit variability. Students will learn about the use of linear models to build statistical descriptions of data and about the use of simple probability distributions to provide useful models for many applications. Students will be able to use regression analysis in some simple cases; they will be able to apply Poisson Processes to model relevant random processes

Module summary

Statistics aims to learn about populations on the basis of samples drawn from them. Population parameters, such as means, can be estimated by suitable sample statistics, but they will be in error because of sample to sample variation. Statistical inference is concerned both with estimating parameters and also with quantifying the associated sampling variation.
This module introduces fundamental notions of a standard error, confidence interval and hypothesis tests, in the context of both discrete and continuous variables. The likelihood is probably the most important concept in statistical methodology, and its introduction in the case of a scalar parameter is one of the main features of the module.

The module will also introduce students to two distinct areas of stochastic modelling. One part will broaden the students' knowledge of statistical inference by introducing the linear model. This will start with a simple regression for a scalar covariate, moving to an introductory treatment of a matrix-based approach for a model with more covariates. The other part will be to use the Poisson process as an example of a model for a process of events occurring randomly in time. The main properties of the homogenous Poisson process will be derived, and necessary tools, such as probability and moment generating functions, will be studied.

Outline Of Syllabus

Notion of an estimator: examples using means and proportions.
Properties of sampling distributions, including standard errors and confidence intervals.
Simulation of a sampling distribution.
Central Limit Theorem via simulation, no proof.
Likelihood, maximum likelihood estimators (scalar case) and their properties, including the illustration of these using simulation. Likelihood ratio test. Sufficiency.
Introduction to hypothesis tests: rejection regions, type I and type II errors, power and significance level. Most powerful tests via the Neyman-Pearson Lemma. Illustrations using one- and two-sample hypothesis tests for means and for proportions.

Simple linear regression, i.e. E(Y) = ?0 + ?1 x for scalar x with Normal errors with unknown variance. Equivalence of least squares and maximum likelihood. Properties of estimator of ? = (?0 ?1). Introduction of the general linear model using matrix formulation; demonstration of formula for the estimator of ?; use of formula for the variance of the estimator of ? but no proof. Examples using regression with two or three continuous covariates.

Introduction to PGFs and MGFs. Formal definition of the homogenous Poisson Process. Distribution of number of events in an interval. Distribution of inter-arrival times and time to nth event. Reinforcement of results using simulation.

Teaching Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture401:0040:00Formal Lectures – Present in Person
Scheduled Learning And Teaching ActivitiesLecture41:004:00Revision Lectures – Present in Person
Scheduled Learning And Teaching ActivitiesLecture101:0010:00Problem Classes – Synchronous On-Line
Guided Independent StudyAssessment preparation and completion1061:00106:00Preparation time for lectures, background reading, coursework review
Scheduled Learning And Teaching ActivitiesDrop-in/surgery101:0010:00Synchronous On-Line
Guided Independent StudyIndependent study301:0030:00Completion of in course assessments
Total200:00
Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Methods

The format of resits will be determined by the Board of Examiners

Exams
Description Length Semester When Set Percentage Comment
Written Examination1502A80N/A
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises1M10Problem-solving exercises
Prob solv exercises2M10Problem-solving exercises
Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

In the event of on-campus examinations not being possible, an on-line alternative assessment will be used for written examination 1.

Reading Lists

Timetable