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The next seminar in the first semester seminar programme of the Manchester Centre for Statistical Science will take place in: Room 1.09 The Mathematics Buiding The University of Manchester Oxford Road Manchseter at 2.15 p.m. on Wednesday 11th November. The speaker is: Professor Murray Aitkin, Department of Statistics, University of Newcastle-upon-Tyne. Title: A COMPARISON OF KERNEL AND MIXTURE MAXIMUM LIKELIHOOD DENSITY ESTIMATES Abstract: Kernel methods are widely used in density estimation. An important issue in their construction is the choice of bandwidth. Different choices of bandwidth are advocated in different references, and this remains an area of active research. The S-plus kernel function provides four possible bandwidths. When a Gaussian kernel is used, the kernel estimate is an n-component normal mixture, with means located at the data points and equal masses of 1/n for each component. This formulation makes clear the difficulty of the bandwidth choice: the standard deviation cannot be estimated by maximum likelihood in such a mixture. The above mixture representation suggests an alternative method of estimation of the unknown density: by representing it as a normal density mixed over the mean, nonparametric maximum likelihood can be used to fit the model, as a finite mixture of normals. Within the mixture framework, the choice of the number of components is a difficult theoretical problem. We report a comparison of the kernel and mixture approaches on a number of standard densities, which are themselves generated as finite mixtures of normals. We construct the following estimates: a) Kernel densities using each of the four bandwidths provided in S- plus; b) Mixture densities using both the nonparametric maximum likelihood estimate of the mixing distribution, and a parsimonious estimate using the likelihood ratio test. These estimates are compared by simulation, using the integrated squared error computed directly from each simulation. The comparisons of the methods, both across and within the two approaches, give quite different results for estimating densities which are generated as mixtures of normals with the same variance, and with different variances. The seminar will be followed by tea. All those who are interested are very welcome to attend. Please note that full details of the first semester seminars are available at: http://www.ma.man.ac.uk/News/seminar.htm Peter Foster Dept of Mathematics The University of Manchester %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%