Semiparametric Bayesian information criterion for model selection in ultra-high dimensional additive models

For linear models with a diverging number of parameters, it has recently been shown that modified versions of Bayesian information criterion (BIC) can identify the true model consistently. However, in many cases there is little justification that the effects of the covariates are actually linear. Thus a semiparametric model, such as the additive model studied here, is a viable alternative. We demonstrate that theoretical results on the consistency of the BIC-type criterion can be extended to this more challenging situation, with dimension diverging exponentially fast with sample size. Besides, the assumptions on the distribution of the noises are relaxed in our theoretical studies. These efforts significantly enlarge the applicability of the criterion to a more general class of models.