Detailed course outline

Problem sheet 1 (exponential families, sufficient statistics)

Problem sheet 2 (MLEs, sufficient statistics, bias)

Problem sheet 3 (sufficiency, bias, Rao-Blackwell theorem)

Problem sheet 4 (Rao-Blackwell theorem, Cramer-Rao lower bound)

Problem sheet 5 (Cramer-Rao lower bound, minimum variance unbiased estimators)

Problem sheet 6 (MLEs, functional invariance)

Problem sheet 7 (Neyman-Pearson lemma, UMP tests, monotone likelihood ratio tests)

Problem sheet 8 (monotone likelihood ratio tests, generalised likelihood ratio tests)

Chapter 1 Exponential families, sufficiency, factorisation theorem, invariance of MLEs.

Chapter 2 Rao-Blackwell theorem, Cramer-Rao lower bound, efficiency, minimum variance unbiased estimators.

Chapter 3 Asymptotic theory of Maximum Likelihood Estimators.

Chapter 4 Review of hypothesis testing and the Neyman-Pearson lemma, simple hypotheses, composite alternatives (one and two sided).

Chapter 5 Monotone likelihood ratio tests; Generalised likelihood ratio tests.

A quick note on Chebychev's inequality

Extra for Chapter 3 A sketch proof of the asymptotic distribution for MLEs in the univariate case (not examinable).

Two links to some lecture notes with a more formal treatment of the asymptotics (again, just out of interest). Part I. Part II.

**ASSESSMENT** 100% examination.

Exam papers for the last five years can be found on the library's archive.

Bain and Engelhardt Introduction to probability and mathematical statistics 512.75 BAI

Cox and Hinkley Theoretical statistics 512.76 COX

Hogg and Craig Introduction to mathematical statistics 512.75 HOG

Silvey Statistical inference 512.76 SIL

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