Tuning Parameter Selection in High Dimensional Penalized Likelihood

Abstract

Summary: Determining how to select the tuning parameter appropriately 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 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 the generlized information criterion. We find that this model complexity penalty should diverge at the rate of some power of log (p) depending on the tail probability behaviour of the response variables. This reveals that using the Akaike information criterion or Bayes information criterion to select the tuning parameter may not be adequate for consistently identifying the true model. On the basis of our theoretical study, we propose a uniform choice of the model complexity penalty and show that the approach proposed consistently identifies the true model among candidate models with asymptotic probability 1. We justify the performance of the procedure proposed by numerical simulations and a gene expression data analysis.