  The paper proposes a new covariance estimator for large covariance matrices
when the variables have a natural ordering. Using the Cholesky decomposition of
the inverse, we impose a banded structure on the Cholesky factor, and select
the bandwidth adaptively for each row of the Cholesky factor, using a novel
penalty we call nested Lasso. This structure has more flexibility than regular
banding, but, unlike regular Lasso applied to the entries of the Cholesky
factor, results in a sparse estimator for the inverse of the covariance matrix.
An iterative algorithm for solving the optimization problem is developed. The
estimator is compared to a number of other covariance estimators and is shown
to do best, both in simulations and on a real data example. Simulations show
that the margin by which the estimator outperforms its competitors tends to
increase with dimension.
