CS - Prof Oliver Jenkinson

Optimal Statistical Dispersion 

Supervisor: Prof Oliver Jenkinson

Project description:

Statistical dispersion quantifies the variability of data relative to a central tendency, most commonly the mean, with variance the most usual measure of this. More generally, dispersion metrics are non-negative and increase with data heterogeneity. The aim of this project is to explore dispersion within constrained time series, focusing on distributions that minimize statistical spread under structural or dynamical constraints. 

Beyond classical notions of dispersion, the project will investigate majorization theory, a partial ordering of distributions rooted in economics (Dalton, Lorenz, Rothschild & Stiglitz) and formalised statistically by Blackwell, in the context of comparison of experiments, and Strassen (see also e.g. Torgerson, Whitt). 

Majorization provides a framework for comparing distributions in terms of inequality and leads naturally to stochastic dominance - specifically second-order dominance, which incorporates both mean and variance considerations. 

The project will analyse the structure and implications of first-, second-, and potentially third-order stochastic dominance (in which the coincidence of both mean and variance is necessary for comparability) within empirical and simulated datasets. This includes identifying dominance relations among distributions generated by time series models, and characterising those with minimal dispersion. 

There will be scope for both theoretical and computational work, with the balance between these approaches being determined by the student's preferences. At an experimental level, the student will be involved in computer experiments aimed at generating and analysing data. Theoretical work will involve proving results characterising minimal and maximal statistical dispersion, and the circumstances under which such optimal distributions exist. A specific application will be to those distributions with smallest variance around the mean. 

References:

• Blackwell, Equivalent comparisons of experiments, Annals of Mathematical Statistics. 
• Blackwell, Comparison of experiments, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. 
• Dalton, The measurement of the inequality of incomes, The Economic Journal. 
• O. Lorenz, Methods of measuring concentration of wealth, Journal of the American Statistical Association. 
• Rothschild & J. Stiglitz, Increasing risk I: A definition, Journal of Economic Theory. 
• Strassen, The existence of probability measures with given marginals, Annals of Mathematical Statistics. 
• Torgerson, Comparison of statistical experiments, Cambridge University Press. 
• Whitt, Bivariate distributions with given marginals, Annals of Statistics.

Funding Notes:

Further information: 
How to apply 
Entry requirements 
Fees and funding

As one of the UK’s most diverse universities, QMUL fosters an inclusive and supportive academic community.

The School of Mathematical Sciences is committed to the equality of opportunities and to advancing women’s careers. As holders of a Bronze Athena SWAN award, we offer family-friendly benefits and support part-time study.