A change point problem occurs in many statistical applications. If there
exist change points in a model, it is harmful to make a statistical analysis
without any consideration of the existence of the change points and the results
derived from such an analysis may be misleading. There are rich literatures on
change point detection. Although many methods have been proposed for detecting
multiple change points, using these methods to find multiple change points in a
large sample seems not feasible.
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) and the adaptive LASSO
penalty (Zou, 2006) have been proposed in literature. In this article, we
establish the result that using Bayesian information criterion (BIC) to select
the tuning parameter in penalized likelihood estimation with both types of
penalties can lead to consistent graphical model selection.