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.