Penalized Likelihood Density Estimation: Direct Cross-Validation and Scalable Approximation

For smoothing parameter selection in penalized likelihood density estimation, a direct crossvalidation strategy is illustrated. The strategy is as effective as the indirect cross-validation developed earlier, but is much easier to implement in multivariate settings. Also studied is the practical implementation of certain low-dimensional approximations of the estimate, with the dimension of the model space selected to achieve both asymptotic efficiency and numerical scalability. The greatly reduced computational burden allows the routine use of the technique for the analysis of large data sets. Related practical issues concerning multivariate numerical integration are also briefly addressed.