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Theory of Inference (MATH 35600)

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Administrative Information

  1. Unit number and title: MATH 35600 Theory of Inference
  2. Level: H/6
  3. Credit point value: 10 credit points
  4. Year: 10/11
  5. First Given in this form: 2000-01
  6. Unit Organiser: Stanislav Volkov
  7. Lecturer: Dr Stanislav Volkov
  8. Teaching block: 2
  9. Prerequisites: MATH11300 Probability 1 and MATH 11400 Statistics 1.

General Description of the Unit

Statistical inference is the science concerned with drawing inferences on the basis of uncertain data. In contrast to numerical or graphical descriptive techniques, which have the relatively simple aim of summarising the data actually observed, in inference we intend to draw conclusions about the populations from which the data are drawn. In doing so, almost universally, a probabilistic model is built for the mechanism generating the data, and the specific objects of inference are the unknowns (parameters) appearing in such models. There are several approaches for doing this and we shall cover the main features of the classical (or frequentist inference). One of the aims of the unit is to present this approach to statistical inference from a theoretical perspective, rather than an applied one, focusing on the mathematical properties of the statistical methods.

Relation to Other Units

This unit develops the theoretical background to much of the statistical methodology covered in statistics units at levels 1, 2 and 3.

Teaching Methods

Lectures, exercises to be done by students.

Learning Objectives

To gain an understanding of:

  • General objectives of statistical inference;
  • Types of inference: point and interval estimation, hypothesis testing, and associated optimality criteria

Assessment Methods

The final assessment mark is calculated from a 1½-hour written examination in May/June consisting of THREE questions. A candidate's best TWO answers will be used for assessment. Calculators are NOT permitted for this examination.

Award of Credit Points

Either : a final assessment mark of 40 or more, 
or reasonable attempts at specified written work during the course together with a final assessment mark of 30 or more.

Transferable Skills

Self-assessment by working examples sheets and using solutions provided.

Texts

There is no set textbook for the unit. However, the following texts cover much of the relevant material and may provide useful background reading.

  1. G. Casella and R. L. Berger, Statistical Inference, Duxbury Press, 2001.
  2. E. L. Lehmann, Testing Statistical Hypotheses, Springer, 1997.
  3. E. L. Lehmann and G. Casella, Theory of Point Estimation, Springer, 2003.
  4. P. J. Bickel and K. A. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics, Vol I, Prentice Hall, 2000.
  5. D.A.S. Fraser, Probability and Statistics; Theory and Applications, ITS, 1991

Syllabus

  1. Sufficiency, completeness, ancillarity, regular exponential families, Fisher Information.
  2. Point estimation: Cramer-Rao lower bound, Rao-Blackwell theorem, UMVU estimators, efficiency.
  3. Hypothesis testing: power and size, Neyman-Pearson Lemma, uniformly most powerful tests.
  4. Set estimation: pivotal quantities, relationship between hypothesis tests and set estimators.