Some Properties On Extended Bayes Information Criterion

It is well known that there are two popular types of model selection methods in literature. One type selects variables via various shrinkage methods, such as least absolute shrinkage and selection operator (LASSO), smoothly clipped absolute de-viation (SCAD) and minimax concave penalty (MCP), etc., among which SCAD and MCP can also select the true model consistently. The other one is the best sub-set procedure, such as, Akaike’s information criterion (AIC), cross-validation (CV), generalized cross-validation (GCV), risk inflation criterion (RIC), Bayes information criterion (BIC), extended Bayes information criterion (EBIC), etc., among which EBIC can also select the true model consistently.In this thesis, we focus on the issue of consistency for the best subset procedure. For the generalized linear model (GLM), under some regularity conditions, Chen and Chen (2012) demonstrated that the EBIC estimator is n1/3-consistent, and also showed that under some conditions, the EBIC can select the true model consistently, while the RIC can not select the true model. In this thesis, we will show that the EBIC estimator is (?)-consistent, which is better than the results of Chen and Chen (2012). We also show that under an additional condition, RIC can also select the true model consistently.