Bootstrap confidence intervals for conditional density function in Markov processes

Bootstrap confidence intervals for the conditional density function in Markov processes are established. For this purpose, recent bootstrap algorithms constructed for prediction intervals are reviewed and extended to this context. These methods are: model-free bootstrap, predictive model-free, limit model-free, bootstrap for nonparametric autoregressive models with predictive and fitted residuals, local bootstrap and bootstrap based on estimates of the transition density. However, in order to achieve a good coverage probability, the choice of an appropriate smoothing parameter for conditional density estimation turns out to be of utmost importance. In this sense, cross validation and plug-in bandwidth parameter selectors are considered, as well as a deterministic one. An extensive simulation study is carried out to show the empirical behavior of these methods and to compare them. Finally, the methods are illustrated by applying them to a real data set.