MMath Mathematics

This course aims to introduce students to the principles of estimation and hypothesis testing, and to familiarize them with the effective methods for estimation and constructing test procedures.

This course aims to introduce students to the principles of estimation and hypothesis testing, and to familiarize them with the effective methods for estimation and constructing test procedures. 

On the successful completion of the course, students will be able to:  

1. Explain the properties of exponential family and apply these concepts to practical examples.  
2. Formulate estimators using the maximum likelihood principle, and analyse their non-asymptotic and asymptotic properties, with application to the exponential family.
3. Construct likelihood-based confidence intervals for parameters, including their asymptotic forms, and perform hypothesis testing using generalised likelihood ratio tests and related methods.
4. Explain the key concepts of multiple testing, including FWER and FDR, and apply methods such as the Bonferroni correction and Benjamini-Hochberg procedure to control error rates.
5. Apply computational techniques to solve statistical inference problems.  
6. Apply advanced methods such as the EM algorithm and Kaplan-Meier estimator to address incomplete and censored data problems.

Part A:

• Exponential Family: definition and examples, canonical parameters and statistics, dispersion parameter, cumulant functions
• Maximum likelihood estimation (MLE): theoretical properties including asymptotics, restricted MLE, Fisher information, Cramer-Rao inequality, efficiency, most efficient estimators, sufficiency and minimal sufficiency, MLE for exponential family
• Inference: likelihood-based confidence intervals, Wald test, (generalised) likelihood ratio test, asymptotic form of the generalised likelihood ratio test

Part B:

• Multiple testing and simultaneous inference: it involves the control of various types of error rates, such as Familywise Error Rate (FWER), False Discovery Rate (FDR). Other topics include Bonferroni method, Benjamini-Hochberg procedure etc.
• Computational inference: Bootstrap, Monte Carlo and bootstrap tests, Bootstrap confidence intervals, Kernel density estimation (KDE)
• Incomplete data: EM algorithm, Kaplan-Meier estimator for censored survival data 

Teaching is composed of two hours of lectures and one tutorial class per week. Teaching materials will be made available online for reference and review. 

Written exam - 80% weighting

Mid-term - 20% weighting

Generic feedback will be provided after marks are released.  

Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury Press.  

Garthwaite, P. H., Jolliffe, I. T., & Jones, B. (2002). Statistical Inference (2nd ed.). Oxford University Press.

Lauritzen, S. (2023). Fundamentals of mathematical statistics. Taylor Francis.

Abramovich, F. & Ritov, Y. (2023). Statistical theory: a concise introduction (2nd edition). Taylor Francis.  

Davison, A. C., & Hinkley, D. V. (1997). Bootstrap Methods and Their Application. Cambridge University Press.

Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman & Hall.

Hsu, J. (1996). Multiple Comparisons: Theory and Methods. Wiley.

Bretz, F., Hothorn, T., & Westfall, P. (2010). Multiple Comparisons Using R. Springer.