Bayesian approach in the nonparametric conditional density estimation

In the literature, it is well known that in kernel density estimation, plug-in and cross-validation techniques, for the selection of the smoothing parameter, tend to provide under-or over-smoothed estimators when the sample size is small or medium, or the function to be estimated is complex. To overcome this latest problem, recently, the Bayesian approach has been proposed as an alternative to these classical methods. In this paper, we restricted attention to extending the idea of the Bayes rule to estimate the smoothing parameter of the conditional density kernel estimation. We are interested in the impact of the use of the global Bayesian approach for the smoothing parameter selection on the performance (the average of the Integrated Squared Error (ISE) and the calculation time required for their implementation) of the kernel conditional density estimator compared to the classical method trough a Monté Carlo simulation. Moreover, based on simulated samples of different sizes from two different conditional models, we adopted the Bayesian approach compared with the classical smoothing parameter selection procedure (the cross-validation method), using the Gaussian kernel to construct the estimators in question.