Consistent estimator for basis selection based on a proxy of the Kullback-Leibler distance

Given a random sample from a continuous and positive density f, the logistic transformation is applied and a log density estimate is provided by using basis functions approach. The number of basis functions acts as the smoothing parameter and it is estimated by minimizing a penalized proxy of the Kullback-Leibler distance which includes as particular cases AIC and BIC criteria. We prove that this estimator is consistent.