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Online course materials for MATH38001

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



Additional Requirements

Please note

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)


This course unit aims to introduce students to the principles of efficient estimation and hypothesis testing and acquaint them with the more successful methods of estimation and of constructing test procedures.


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

Learning outcomes

On successful completion of this course unit students will be able

  • to determine how good an estimator or test procedure is on a number of criteria;
  • to construct estimators and test procedures based both on the maximum likelihood principle.

Assessment Further Information

End of semester examination: two hours weighting 100%


  • Estimation: point estimation, unbiasedness, mean square error, consistency, sufficiency, factorization theorem, Cramer-Rao inequality, the score function, Fisher information; efficiency: most efficient estimators, minimal sufficiency, Rao Blackwell theorem and its use in improving an estimator. [8]
  • Methods of estimation: maximum likelihood estimators (m.l.e) and their asymptotic properties, asymptotic distribution of the score function. Confidence intervals based on the m.l.e and on the score function (multivariate case included). Restricted m.l.e and their asymptotic properties. [7]
  • Hypothesis testing: Neyman-Pearson criteria, size and power function. Simple null vs simple alternative hypothesis and the Neyman-Pearson lemma. Hypothesis tests based on (i) m.l.e's; (ii) score function; (iii) the generalised likelihood ratio, profile log-likelihood and its use in interval estimation. The Deviance function and graphical methods in obtaining confidence regions for parameters. [9]

Recommended reading

  • 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.
  • Silvey, S. D., Statistical Inference, Chapman & Hall 1075.

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

Yang Han - Unit coordinator

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