  We address the identifiablity and estimation of recursive max-linear
structural equation models represented by an edge weighted directed acyclic
graph (DAG). Such models are generally unidentifiable and we identify the whole
class of DAGs and edge weights corresponding to a given observational
distribution. For estimation, standard likelihood theory cannot be applied
because the corresponding families of distributions are not dominated. Given
the underlying DAG, we present an estimator for the class of edge weights and
show that it can be considered a generalized maximum likelihood estimator. In
addition, we develop a simple method for identifying the structures of the
DAGs. With probability tending to one at an exponential rate with the number of
observations, this method correctly identifies the class of DAGs and,
similarly, exactly identifies the possible edge weights.
