We strongly encourage applications from strong candidates interested in pursuing a PhD in probability or statistics. Below are some suggested projects that are on offer; we also warmly welcome suggestions for other projects that you might be interested in, and which have connections with our research interests.

Dr Stephen Connor

• Coupling of Markov chains
• Statistical network analysis

Dr Degui Li

Please contact Dr Li directly if you are interested in pursuing a PhD with him.

Dr Samer Kharroubi

• Statistical methods in health economics

Dr Sonia Mazzi

 Please contact Dr Mazzi directly if you are interested in pursuing a PhD with her.

Prof. Wenyang Zhang

Prof. Wenyang Zhang is interested in taking on PhD students to work on Nonparametric Statistics, Nonlinear Time Series, Survival Analysis, Functional Data Analysis, Spatial Data Analysis, Multi-level Modelling, Structural Equation Models. The following is one example of the PhD projects he would like to supervise. Please contact him directly for more PhD projects.

• Estimation of Time Varying High Dimensional Covariance Matrix and Its Application in Optimal Portfolio Allocation in Finance.

• Coupling of Markov chains
  

-- Dr Stephen Connor

Let X and Y be two copies of a Markov chain, begun from different states. We are often interested in constructing X and Y so that if they meet one another then they remain together from then onwards - such a construction is known as a coupling. Couplings are interesting for a number of reasons, not least because they can provide information about the rate at which X converges to its stationary distribution (if it has one). They also have a number of practical uses, in particular in the construction of perfect simulation algorithms. There are lots of interesting open problems in coupling theory, many of which would make good PhD projects for students interested in stochastic processes.

• Statistical network analysis
  

-- Dr Stephen Connor, with Dr Dan Franks (YCCSA) and Dr Jon Pitchford (YCCSA)

There has been a rapid growth of interest in the science of networks, from a wide range of disciplines. There are, however, some significant mathematical problems. The foundation on which network theory is reliant is currently incomplete: in particular, there is a lack of formal theory regarding weighted networks and dynamic networks. This is a serious problem given that almost all real-world networks are both dynamic and weighted. For example, in a social network people will be connected to different degrees (e.g. strength of friendship) and this can change over time. The development of models that consider dynamically changing weighted networks will allow us to build an understanding of such real-world systems. There is also a serious lack of statistical methods for modelling and analyzing empirical network datasets: for example, there is currently no satisfactory way to statistically compare networks that differ in the number of nodes or number of connections. This project will aim to address these problems using modern mathematical, computational and statistical techniques, motivated by examples from social networks and systems biology.

• Statistical methods in health economics

-- Dr Samer Kharroubi

The main concern of the field of health economics is to examine the cost-effectiveness of medical technologies. Whenever new drugs, treatments or medical devices (collectively called medical technologies) are proposed, their sponsors have to demonstrate that they are safe and effective before they can be licensed for public use. Demonstration of safety and efficacy is generally done by conducting a substantial clinical trial, followed by statistical analysis of the data. This is a well-developed area of medical statistics and so a fruitful area for ongoing research.

• Estimation of Time Varying High Dimensional Covariance Matrix and Its Application in Optimal Portfolio Allocation in Finance

-- Prof. Wenyang Zhang

This project is stimulated by the optimal portfolio allocation in finance. It is well know that the optimal portfolio allocation can be expressed in terms of the covariance matrix of the returns of the stocks under consideration. The estimation of covariance matrix would not be a big deal when sample size is much larger than the size of the covariance matrix, for example, the sample covariance matrix would be a good estimator. However, when the sample size and the size of the covariance matrix are comparable, which is the case in portfolio allocation as the number of stocks is typically of the same order as the sample size, the sample covariance matrix as an estimator of the covariance matrix would run into trouble. This is because the estimated optimal portfolio allocation depends on the inverse of the estimator of the covariance matrix, when the size of the covariance is large, the random errors of the estimators of the elements in the covariance matrix will accumulate, which will make the estimator of the optimal portfolio allocation very poor.

Most existing literature about the estimation of the covariance matrix is based on the assumption that the covariance matrix is a constant matrix. This is clearly not a realistic assumption in portfolio allocation as today's optimal portfolio allocation may not be optimal next month. What is more realistic is to assume the optimal portfolio allocation depends on time and estimate it through the estimated time-varying covariance matrix of the returns of the stocks.

This project is going to investigate the dynamic structure of the time-varying covariance matrix of large size and construct the estimation procedure for the time-varying covariance matrix and the time-varying optimal portfolio allocation. In summary, this project is going to develop a new estimation procedure for time-varying covariance matrix of large size, establish asymptotic properties to justify the estimation method, and apply the method to analyse some financial data sets from London Stock Market and Shanghai Stock Market.