  The statistical censoring setup is extended to the situation when random
measures can be assigned to the realization of datapoints, leading to a new way
of incorporating expert information into the usual parametric estimation
procedures. The asymptotic theory is provided for the resulting estimators, and
some special cases of practical relevance are studied in more detail. Although
the proposed framework mathematically generalizes censoring and coarsening at
random, and borrows techniques from M-estimation theory, it provides a novel
and transparent methodology which enjoys significant practical applicability in
situations where expert information is present. The potential of the approach
is illustrated by a concrete actuarial application of tail parameter estimation
for a heavy-tailed MTPL dataset with limited available expert information.
