Bernstein estimator for unbounded density function

Nonparametric estimation for an unknown probability density function f with a known compact support [0, 1] not necessarily bounded at x=0 is considered. For such class of density functions, we consider the Bernstein estimator. The uniform weak consistency and the uniform strong consistency on each compact I in (0, 1) are established for the Bernstein estimator. We prove also the almost sure convergence to infinity at x=0 of the Bernstein estimator when the density function f is unbounded at x=0. To select the optimal bandwidth parameter of the Bernstein estimator, the least squares cross-validation and the likelihood cross-validation methods are developed.