## Statistical Inference

 Unit code: MATH38001 Credit Rating: 10 Unit level: Level 3 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

#### Requisites

Prerequisite

Students are not permitted to take more than one of MATH38001 or MATH48001 for credit in the same or different undergraduate year.

Students are not permitted to take MATH48001 and MATH68001 for credit in an undergraduate programme and then a postgraduate programme.

Note that MATH68001 is an example of an enhanced level 3 module as it includes all the material from MATH38001

When a student has taken level 3 modules which are enhanced to produce level 6 modules on an MSc programme taken within the School of Mathematics, then they are limited to a maximum of two such modules (with no alternative arrangements available otherwise)

#### Aims

• To introduce students to the fundamental but important topics in statistical sciences, including the principles of efficient estimation and hypothesis testing.
• To familiarise students with the theories and methods of statistical inference.

#### Overview

Statistical Inference is the body of principles and methods underlying the statistical analysis of data. In this course we introduce desirable properties that good parameter estimates and hypothesis tests should enjoy and use them as criteria in the development of optimal parameter estimators and test procedures.

#### Learning outcomes

On successful completion of this course unit students will be able

• explain key ideas underpinning likelihood-based inference and Bayesian analysis,
• describe how good an estimator or test procedure is on a number of criteria,
• formulate estimators and test procedures based on the maximum likelihood principle,
• use the learnt statistical methods to analyse certain real life data.

#### Assessment Further Information

End of semester examination: two hours weighting 100%

#### Syllabus

• Frequentist and Bayesian approaches, likelihood function; prior and posterior distributions; Bayesian rule, point estimation; unbiasedness; mean squared error; consistency; score function; Fisher information; Cramer-Rao inequality; efficiency; most efficient estimators; sufficiency; factorisation theorem; minimal sufficiency. [8]
• Methods of estimation: maximum likelihood estimators (MLE) and their asymptotic properties including consistency and asymptotic normality; asymptotic distribution of the score function; confidence intervals based on the MLE and the score function; restricted MLE and its properties. [8]
• Hypothesis testing: hypothesis, types of error; power; Wald test; generalised likelihood ratio test; asymptotic form of the generalised likelihood ratio test; multinomial test; Pearson Chi-squared statistic. [6]

• Beaumont, G. P., Intermediate Mathematical Statistics. Chapman & Hall 1980.
• Cox, D. R. and Hinkley, D. V., Theoretical Statistics. , Chapman & Hall 1974.
• Lindgren, B. W. Statistical Theory, 4th edition, Chapman & Hall 1993.
• Mood, A. M., Graybill, F. A. and Boes, D. C., Introduction to the Theory of Statistics, 3rd edition, McGraw-Hill 1974.
• Pawitan, Y., In All Likelihood: Statistical Modelling and Inference Using Likelihood. Oxford University Press, 2001.

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

#### Teaching staff

Jianxin Pan - Unit coordinator