Unit information: Statistical Inference in 2016/17

Unit name	Statistical Inference
Unit code	MATHM6009
Credit points	10
Level of study	M/7
Teaching block(s)	Academic Year (weeks 1 - 52)
Unit director	Dr. Kovac
Open unit status	Not open
Pre-requisites	None
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

This unit will cover the principles and systematic structure of statistical inference, in a way that enables students to understand the uncertainty involved in the conclusions of statistical analyses, and assess the relative merits of different frameworks for statistical analysis. The unit is aimed at students who already have a basic knowledge of main components of statistical inference but wish to develop a deeper understanding of the relationship of these elements within different inference paradigms. Key ideas about probability models and the objectives of statistical analysis are introduced and the differences between the Bayesian and frequentist analyses are illustrated. The topics covered may include: Likelihood, sufficiency, ancillarity, conditionality and the fundamentals of exponential families. Statistical decision theory, minimax and Bayes rules, admissibility, Stein's paradox, hypothesis testing as a decision problem. Subjective and frequency interpretation of probability, the Bayesian paradigm, DeFinetti's theorem, conjugate and reference priors, empirical Bayes and related methods. Model comparison, significance testing and structural uncertainty.

Aims:

The aim of this unit is to provide the students with a solid understanding of the main paradigms of statistical inference, their strengths and limitations, their similarities and differences, and their role in underpinning statistical methodology.

Only available as part of a 1+ 3 Statistics MRes + PhD programme.

Intended Learning Outcomes

To be able to:

• Identify an appropriate choice of model and formulate appropriate objectives for simple standard problems involving interval estimation, tests of significance, prediction and decision theory, and justify their choices in each case.

• Compare model formulations in Bayesian and frequentist frameworks, and the role and interpretation of parameters in each case.

• Describe the concepts of likelihood, sufficiency (and sufficient statistics) and exponential families, explain their role in identifying appropriate prior distributions and pivotal quantities, and compute relevant quantities for standard examples.

• Compare and contrast the underlying principles and methods of frequentist and Bayesian frameworks for significance and hypothesis tests, perform appropriate tests for standard models, and compare and contrast the interpretation of the outcomes in the different frameworks.

• Compare the interpretations of uncertainty in frequentist and Bayesian frameworks

Teaching Information

Lectures, problem sheets and tutorials.

Assessment Information

Assessment will be based on an extended assignment, discussing fundamental aspects of inference in a comparative way based on one or more specific examples, possibly taken from the literature.

The assessment criteria for the assignment will be based on a suitably modified version of the current Mathematics Department Project Assessment form. The assignment will be marked by the member of staff in charge of the unit and by an independent second marker.

Reading and References

• Principles of Statistical Inference, D R Cox, CUP, 2006

• Comparative Statistical Inference, V Barnett & V D Barnett, Wiley, 1999

• Bayesian Theory, J M Bernardo & A F M Smith, Wiley, 1994

• Theory of Point Estimation, E L Lehman & G C Casella, Springer, 2001

• Testing of Statistical Hypotheses, E L Lehman & J P Romano, Springer, 2004

• Empirical Likelihood, A B Owen, Chapman & Hall/CRC, 2001