## Abstract

Graphical models are frequently used to explore networks, such as genetic networks, among a set of variables. This is usually carried out via exploring the sparsity of the precision matrix of the variables under consideration. Penalized likelihood methods are often used in such explorations. Yet, positive-definiteness constraints of precision matrices make the optimization problem challenging. We introduce nonconcave penalties and the adaptive LASSO penalty to attenuate the bias problem in the network estimation. Through the local linear approximation to the nonconcave penalty functions, the problem of precision matrix estimation is recast as a sequence of penalized likelihood problems with a weighted L _{1} penalty and solved using the efficient algorithm of Friedman et al. [Biostatistics 9 (2008) 432-441]. Our estimation schemes are applied to two real datasets. Simulation experiments and asymptotic theory are used to justify our proposed methods.

Original language | English (US) |
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Pages (from-to) | 521-541 |

Number of pages | 21 |

Journal | Annals of Applied Statistics |

Volume | 3 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2009 |

## Keywords

- Adaptive LASSO
- Covariance selection
- Gaussian concentration graphical model
- Genetic network
- LASSO
- Precision matrix
- SCAD

## ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Statistics, Probability and Uncertainty