A Review and Some New Proposals for Bandwidth Selection in Nonparametric Density Estimation for Dependent Data

When assuming independence, the choice of the smoothing parameter in density estimation has been extensively studied. However, when considering dependence, very few papers have dealt with data-driven bandwidth selectors. First of all, we review the state of art of the existing methods for bandwidth selection under dependence. Moreover, three new (or adapted) methods are proposed: (a) an extension to the dependent case of the modified cross-validation by Stute (J Statisti Plann Infer 30:293–305, 1992; b) an adaptation to density estimation of the penalized cross-validation proposed by Estévez-Pérez et al. (J Statisti Plann Infer 104:1–30, 2002) for hazard rate estimation; (c) finally, the smoothed version of the so-called moving blocks bootstrap is established and an exact expression for the bootstrap version of the mean integrated squared error under dependence is obtained in this context. This is useful since Monte Carlo approximation is not needed to implement the bootstrap selector. An extensive simulation study is carried out in order to check and compare the empirical behaviour of six selected bandwidths.