Silverman, B. W.; Jones, M. C.; Wilson, J. D. and Nychka, D. W
(1990).
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, 52(2) pp. 271–324.

URL: | http://www.jstor.org/stable/2345438 |
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Google Scholar: | Look up in Google Scholar |

## Abstract

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.

Item Type: | Journal Article |
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Copyright Holders: | 1990 Royal Statistical Society |

ISSN: | 1467-9868 |

Keywords: | ill-posed problems; indirect observations; intensity estimation; maximum likelihood; penalized likelihood; positron emission tomography; smoothing; stereology |

Academic Unit/Department: | Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics Faculty of Science, Technology, Engineering and Mathematics (STEM) |

Item ID: | 28328 |

Depositing User: | Sarah Frain |

Date Deposited: | 16 Mar 2011 10:01 |

Last Modified: | 04 Oct 2016 11:01 |

URI: | http://oro.open.ac.uk/id/eprint/28328 |

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