Research in Nonparamentric Smoothing
Non-parametric smoothing methodology, particularly for density and regression functions consists of techniques that make few assumptions about the form of an underlying function that needs to be estimated. Essentially these techniques allow the data to "speak for itself" in determining the final estimate which is constructed by weighted local averaging, the extent of which is controlled by a smoothing parameter or bandwidth. However, these estimators are biased and so a loss function which comprises both squared bias and variance is commonly used. Integrating the loss function over the full range space results in the mean integrated squared error (MISE) as a global criterion which can be used as a means of comparing the (theoretical) performance of different estimators. More practically, for a given dataset, an estimate of the MISE can be optimised to determine, for example, the size of the smoothing parameter to be used in the function estimate. For an excellent introduction to this area see B.W. Silverman (1986) "Density Estimation for Statistics and Data Analysis" Chapman & Hall, London.
In his work, Dr Peter Foster has investigated estimators with better properties (bias, MISE) than standard approaches such as those based on fixed kernels, and their applications in multivariate statistics. A new and effective boundary correction technique was considered which can be used in kernel density estimation. Adaptive kernel estimators and use of bootstrap techniques for and width selection were also considered. Statistical applications include testing for multivariate normality, assessing the multimodality of a multivariate distribution and a new approach to the exploratory data analysis of multivariate data based on identifying directions of high density. Plans for future research include further investigation of adaptive estimators, including smoothing parameter choice and their application in various statistical techniques, bias reduction in a multivariate setting as well as a more detailed look at the general effectiveness and applications of multivariate estimators. Dr Foster also has an ongoing collaboration with a research group at the Manchester Royal Children's Hospital which has resulted in a number of publications looking at various aspects of childhood growth. This work is continuing and involves longitudinal data analysis and applications of smoothing techniques to this area.