  Recursive max-linear vectors model causal dependence between node variables
by a structural equation model, expressing each node variable as a max-linear
function of its parental nodes in a directed acyclic graph (DAG) and some
exogenous innovation. For such a model, there exists a unique minimum DAG,
represented by the Kleene star matrix of its edge weight matrix, which
identifies the model and can be estimated. For a more realistic statistical
modeling we introduce some random observational noise. A probabilistic analysis
of this new noisy model reveals that the unique minimum DAG representing the
distribution of the non-noisy model remains unchanged and identifiable.
Moreover, the distribution of the minimum ratio estimators of the model
parameters at their left limits are completely determined by the distribution
of the noise variables up to a positive constant. Under a regular variation
condition on the noise variables we prove that the estimated Kleene star matrix
converges to a matrix of independent Weibull entries after proper centering and
scaling.
