# PhD projects

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 Degui Li

Dr Samer Kharroubi

Dr Marina Knight

Dr Sonia Mazzi

Dr Andrea Wiencierz

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.

• ### Statistical methods in health economics

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

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.

## PhD projects

Overview

Projects in individual research areas: