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# MATH3044 Statistical Inference

## Module Overview

Statistical inference involves using data from a sample to draw conclusions about a wider population. Given a partly specified statistical model, in which at least one parameter is unknown, and some observations for which the model is valid, it is possible to draw inferences about the unknown parameters and hence about the population from which the sample is drawn. As such, inference underpins all aspects of statistics. However, inference can take different forms. It may be adequate to provide a point estimate of a parameter, i.e. a single number. More usually, an interval is required, giving a measure of precision. It may also be necessary to test a pre-specified hypothesis about the parameter(s). These forms of inference can all be considered as special cases of the use of a decision function. There are a number of different philosophies about how these inferences should be drawn, ranging from that which says the sample contains all the information available about a parameter (likelihood), through that which says account should be taken of what would happen in repeated sampling (frequentist), to that which allows the sample to modify prior beliefs about a parameter’s value (Bayesian). This Module aims to explore these approaches to parametric statistical inference, particularly through application of the methods to numerous examples.

### Aims and Objectives

#### Module Aims

Derive suitable point estimators of the parameters of the distribution of a random variable and give a measure of their precision

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Derive suitable point estimators of the parameters of the distribution of a random variable and give a measure of their precision
• Use computational methods to obtain and evaluate estimators
• Appreciate the differences between inference paradigms and how they can be embedded in decision theory
• Demonstrate knowledge and understanding of: Test a hypothesis concerning the distribution of a random variable; Evaluate different estimators using their theoretical properties; Understand how to update the prior distribution to obtain the posterior distribution in a Bayesian analysis, and be able to apply this knowledge to simple conjugate analyses

### Syllabus

Point Estimation Sufficiency and the factorisation theorem Maximum Likelihood Estimation (MLE) Unbiased estimation Sufficiency Rao-Blackwell theorem Cramer-Rao lower bound Minimum Variance Unbiased Estimators (MVUE) Asymptotic efficiency Hypothesis Testing Neyman-Pearson Lemma Uniformly most powerful test Likelihood ratio test Computational Inference Numerical solutions to Maximum Likelihood Estimation – Newton-Raphson and Fisher Scoring Re-sampling methods – Jacknife and Bootstrap Bayesian Inference Bayes theorem Prior and posterior distributions Uniform and conjugate prior distributions Predictive inference Decision-Based Inference Loss functions and risk functions Minimax decisions Admissibility Bayes risk

### Learning and Teaching

#### Teaching and learning methods

Lectures, problem classes, coursework, exercises, private study

TypeHours
Lecture36
Tutorial12
Independent Study102
Total study time150

#### Resources & Reading list

Mukhopadhyay H (2006). Introductory Statistical Inference.

Garthwaite OH, Jolliffe IT & Jones B (2002). Statistical Inference.

Lee PM (2004). Bayesian Statistics : An Introduction.

Young GA & Smith RL (2005). Essentials of Statistical Inference.

### Assessment

#### Assessment Strategy

80% written exam, 20% coursework Referral arrangements: Written Examination

#### Formative

Assignments and problem sheets

#### Summative

MethodPercentage contribution
Coursework 20%
Exam  (2 hours) 80%

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External

### Linked modules

Pre-requisite: MATH2011

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.

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