# School of Mathematics and Statistics

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2013/2014 Sem. 1

## Aims

- To show how the methods of estimation and hypothesis testing met in MT2004 and MT3606 can be justified and derived.

- To extend those methods to a wider variety of situations.

## Objectives

By the end of the course students are expected to

- understand the main problems in comparing the usefulness of two estimators.

- identify sufficient statistics and appreciate why they are important.

- know how to improve estimators using the Rao-Blackwell Theorem.

- derive bounds on variances provided by the Cramér-Rao inequality.

- derive maximum likelihood estimators of one and two dimensional parameters and establish their major properties.

- apply the Neyman-Pearson lemma for testing simple hypotheses.

- determine and use the generalised likelihood ratio statistic when testing composite hypotheses.

- appreciate the relationship between frequentist confidence sets and tests of hypotheses.

## Syllabus

1. Comparison of Point Estimators:  The framework for parametric inference, Mean Square Error, Unbiased estimators, Sufficiency, Factorisation Theorem, Minimal sufficiency.
2. Distribution Theory:  Conditional distributions and expectations, Central Limit Theorem.
3. Minimum variance unbiased estimation: Rao-Blackwell Theorem, Exponential Families, Lehmann-Scheffé Theorem.
4. Likelihood, Fisher Information and the Cramér-Rao Inequality: The Efficient Score, Fisher Information, Cramér-Rao lower bound, Attainment of the Cramér-Rao lower bound, Multi-dimensional Cramér-Rao inequality.
5. Maximum likelihood estimators: Elementary properties, Consistency and asymptotic efficiency.
6.  Hypothesis Testing:  Definitions, The Neyman-Pearson lemma, Tests of composite hypotheses, Likelihood ratio tests.
7. Confidence Sets:  Relationship with hypothesis tests, pivotal quantities.

## Textbooks

Probability and Statistics, 4th ed.: M H DeGroot & M J Schervish, Pearson;
Statistical Inference: S D Silvey, Chapman & Hall/CRC;
Statistical Inference, 2nd ed.: G Casella & R L Berger, Brooks/Cole;
Mathematical Statistics: K Knight, Chapman & Hall/CRC;
Statistical Theory, 4th ed.: B W Lindgren, Chapman & Hall.

## Assessment

2 Hour Examination = 100%

MT3606

MT5701

## Availability

Academic year 2013/14 in semester 2 at 10

## Lecturer

Dr I B J Goudie

Click here to see the University Course Catalogue entry.

Revised: PMH (December 2013)

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