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Title: Kernel density estimation, Bayesian inference and random effects model
Author: Chan, Karen Pui-Shan
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 1990
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This thesis contains results of a study in kernel density estimation, Bayesian inference and random effects models, with application to forensic problems. Estimation of the Bayes' factor in a forensic science problem involved the derivation of predictive distributions in non-standard situations. The distribution of the values of a characteristic of interest among different items in forensic science problems is often non-Normal. Background, or training, data were available to assist in the estimation of the distribution for measurements on cat and dog hairs. An informative prior, based on the kernel method of density estimation, was used to derive the appropriate predictive distributions. The training data may be considered to be derived from a random effects model. This was taken into consideration in modelling the Bayes' factor. The usual assumption of the random factor being Normally distributed is unrealistic, so a kernel density estimate was used as the distribution of the unknown random factor. Two kernel methods were employed: the ordinary and adaptive kernel methods. The adaptive kernel method allowed for the longer tail, where little information was available. Formulae for the Bayes' factor in a forensic science context were derived assuming the training data were grouped or not grouped (for example, hairs from one cat would be thought of as belonging to the same group), and that the within-group variance was or was not known. The Bayes' factor, assuming known within-group variance, for the training data, grouped or not grouped, was extended to the multivariate case. The method was applied to a practical example in a bivariate situation. Similar modelling of the Bayes' factor was derived to cope with a particular form of mixture data. Boundary effects were also taken into consideration. Application of kernel density estimation to make inferences about the variance components under the random effects model was studied. Employing the maximum likelihood estimation method, it was shown that the between-group variance and the smoothing parameter in the kernel density estimation were related. They were not identifiable separately. With the smoothing parameter fixed at some predetermined value, the within-and between-group variance estimates from the proposed model were equivalent to the usual ANOVA estimates. Within the Bayesian framework, posterior distribution for the variance components, using various prior distributions for the parameters were derived incorporating kernel density functions. The modes of these posterior distributions were used as estimates for the variance components. A Student-t within a Bayesian framework was derived after introduction of a prior for the smoothing prameter. Two methods of obtaining hyper-parameters for the prior were suggested, both involving empirical Bayes methods. They were a modified leave-one-out maximum likelihood method and a method of moments based on the optimum smoothing parameter determined from Normality assumption.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available