Silverman, B. W.; Jones, M. C.; Wilson, J. D. and Nychka, D. W
A smoothed EM approach to indirect estimation problems, with particular reference to stereology and emission tomography.
Journal of the Royal Statistical Society: Series B (Statistical Methodology), 52(2),
There are many practical problems where the observed data are not drawn directly from the density g of real interest, but rather from another distribution derived from g by the application of an integral operator. The estimation of g then entails both statistical and numerical difficulties. A natural statistical approach is by maximum likelihood, conveniently implemented using the EM algorithm, but this provides unsatisfactory reconstructions of g. In this paper, we modify the maximum likelihood-EM approach by introducing a simple smoothing step at each EM iteration. In our experience, this algorithm converges in relatively few iterations to good estimates of g that do not depend on the choice of starting configuration. Some theoretical background is given that relates this smoothed EM algorithm to a maximum penalized likelihood approach. Two applications are considered in detail. The first is the classical stereology problem of determining particle size distributions from data collected on a plane section through a composite medium. The second concerns the recovery of the structure of a section of the human body from external observations obtained by positron emission tomography; for this problem, we also suggest several technical improvements on existing methodology.
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