Tuning parameter selection in high dimensional penalized likelihood

Determining how to appropriately select the tuning parameter is essential in penalized likelihood methods for high-dimensional data analysis. We examine this problem in the setting of penalized likelihood methods for generalized linear models, where the dimensionality of covariates p is allowed to increase exponentially with the sample size n. We propose to select the tuning parameter by optimizing the generalized information criterion (GIC) with an appropriate model complexity penalty. To ensure that we consistently identify the true model, a range for the model complexity penalty is identified in GIC. We find that this model complexity penalty should diverge at the rate of some power of $\log p$ depending on the tail probability behavior of the response variables. This reveals that using the AIC or BIC to select the tuning parameter may not be adequate for consistently identifying the true model. Based on our theoretical study, we propose a uniform choice of the model complexity penalty and show that the proposed approach consistently identifies the true model among candidate models with asymptotic probability one. We justify the performance of the proposed procedure by numerical simulations and a gene expression data analysis.