Graphical models are frequently used to explore networks among a set of variables. Several methods for estimating sparse graphs have been proposed and their theoretical properties have been explored. There are also several selection criteria to select the optimal estimated models. However, their practical performance has not been studied in detail. In this work, several estimation procedures (glasso, bootstrap glasso, adptive lasso, SCAD, DP-glasso and Huge) and several selection criteria (AIC,

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... BIC, CV, ebic, ric and stars) are compared under various simulation settings, such as different dimensions or sample sizes, different types of data, and different sparsity levels of the true model structures. Then we use several evaluation criteria to compare the optimal estimated models and discuss in detail the superiority and deficiency of each combination of estimating methods and selection criteria. iii Acknowledgment First and foremost, I express herein my deepest gratitude to my co-supervisors Dr. Ivor Cribben and Prof. Dr. Rohana Karunamuni for their guidence and support throughout my graduate studies. After being brought to U of A by Dr. Karunamuni, my life as a graduate student started with a knowledgable and patient mentor. All sorts of difficulties, confusion and challenges happened to me during the last two years but, however, Dr. Karunamuni was the one who always sent me quiet but strong support to lift me up to a new stage. I thank Dr. Karunamuni for any of his continual attention to, interest in and concerns with my coursework and research, thought I have not had a chance to take a course with this excellent professor. At the same time, Dr. Cribben opened the gate of research for me and pointed out a path towards the fascinating unknown. The topic about estimating sparse graphical models has been truly motivating me from the bottom of heart and occupying almost all of my efforts. Not only my degree thesis but also everything exchanged during our meetings boosts the quality of myself. It is doubtlessly a period of happy hours to pour my sweats into the field of statistical sciences with Dr. Cribben, especially in his gentle and considerate manner of mentoring and working.