  Covariance estimation for high-dimensional datasets is a fundamental problem
in modern day statistics with numerous applications. In these high dimensional
datasets, the number of variables p is typically larger than the sample size n.
A popular way of tackling this challenge is to induce sparsity in the
covariance matrix, its inverse or a relevant transformation. In particular,
methods inducing sparsity in the Cholesky pa- rameter of the inverse covariance
matrix can be useful as they are guaranteed to give a positive definite
estimate of the covariance matrix. Also, the estimated sparsity pattern
corresponds to a Directed Acyclic Graph (DAG) model for Gaussian data. In
recent years, two useful penalized likelihood methods for sparse estimation of
this Cholesky parameter (with no restrictions on the sparsity pattern) have
been developed. How- ever, these methods either consider a non-convex
optimization problem which can lead to convergence issues and singular
estimates of the covariance matrix when p > n, or achieve a convex formulation
by placing a strict constraint on the conditional variance parameters. In this
paper, we propose a new penalized likelihood method for sparse estimation of
the inverse covariance Cholesky parameter that aims to overcome some of the
shortcomings of current methods, but retains their respective strengths. We ob-
tain a jointly convex formulation for our objective function, which leads to
convergence guarantees, even when p > n. The approach always leads to a
positive definite and symmetric estimator of the covariance matrix. We
establish high-dimensional estima- tion and graph selection consistency, and
also demonstrate finite sample performance on simulated/real data.
