Statistica Sinica 22 (2012), 1123-1146
TUNING PARAMETER
SELECTION FOR
PENALIZED
LIKELIHOOD ESTIMATION OF GAUSSIAN
GRAPHICAL MODEL
Xin Gao, Daniel Q. Pu,
Yuehua Wu and Hong Xu
York
University
Abstract: In a Gaussian graphical model, the
conditional independence between two variables are characterized by
the corresponding zero entries in the inverse covariance matrix.
Maximum likelihood method using the smoothly clipped absolute
deviation (SCAD) penalty (Fan and Li (2001))
has been
proposed in the literature. In this article, we establish the result
that when 
is fixed, using the Bayesian information criterion
(BIC) to select the tuning parameter in penalized likelihood
estimation with the SCAD penalty can lead to consistent graphical
model selection. When increases with the sample size, a modified
BIC with an extra penalty term is proposed. It can consistently
select the true graphical model under the condition that tends
to infinity and all the true edges are included in a bounded subset.
We compare the empirical performance of BIC with the cross
validation method and demonstrate the advantageous performance of
BIC criterion for sparse graphical models through simulation
studies.
Key words and phrases:
BIC, consistency, cross validation, Gaussian graphical model, model
selection, oracle property, penalized likelihood.