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Seminar on Probability and Statistics Wednesday January 16 2008 Tokyo 122 2:50-4:00 pm
Statistical analysis of fragmentation chains
Marc HOFFMANN Universite Paris-est Marne la vallee Abstract We address statistical inference in self-similar conservative
fragmentation chains, when only observations on the size of the
fragments
below a given threshold are available. (Possibly, the measurement of the
fragments themselves are subject to further systematic experimental
noise.) This framework, introduced by Bertoin and Martinez is motivated
by
mineral crushing in mining industry. We compute upper and lower rates of
estimation for several functionals of the dislocation measure, both in a
semi-parametric and a non-parametric framework. The underlying estimated
object is the step distribution of the random walk associated to a
randomly tagged fragment that evolves along the genealogical tree
representation of the fragmentation process. We establish a formal link
with the statistical problem of estimating the overshoot of the
distribution as the crossing level goes to infinity with the size of the
dataset; in particular the difficulty of the estimation problem in the
non-parametric case is comparable to ill-posed linear inverse problems
of
order 1 in signal denoising.
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