  Estimation of the skeleton of a directed acyclic graph (DAG) is of great
importance for understanding the underlying DAG and causaleffects can be
assessed from the skeleton when the DAG is notidentifiable. We propose a novel
method named PenPC toestimate the skeleton of a high-dimensional DAG by a
two-stepapproach. We first estimate the non-zero entries of a
concentrationmatrix using penalized regression, and then fix the
differencebetween the concentration matrix and the skeleton by evaluating aset
of conditional independence hypotheses. For high dimensionalproblems where the
number of vertices $p$ is in polynomial orexponential scale of sample size $n$,
we study the asymptoticproperty of PenPC on two types of graphs:
traditionalrandom graphs where all the vertices have the same expected numberof
neighbors, and scale-free graphs where a few vertices may have alarge number of
neighbors. As illustrated by extensive simulationsand applications on gene
expression data of cancer patients, PenPChas higher sensitivity and specificity
than the standard-of-the-artmethod, the PC-stable algorithm.
