  We explore statistical inference in self-similar conservative fragmentation
chains when only approximate observations of the sizes of the fragments below a
given threshold are available. This framework, introduced by Bertoin and
Martinez [Adv. Appl. Probab. 37 (2005) 553--570], is motivated by mineral
crushing in the mining industry. The underlying object that can be identified
from the data is the step distribution of the random walk associated with a
randomly tagged fragment that evolves along the genealogical tree
representation of the fragmentation process. We compute upper and lower rates
of estimation in a parametric framework and show that in the nonparametric
case, the difficulty of the estimation is comparable to ill-posed linear
inverse problems of order 1 in signal denoising.
