On the use of pilot estimators in bandwidth selection

Many data based methods for choosing the bandwidth of a kernel density estimator depend on unknown constants associated with auxiliary bandwidths which arise at the functional estimation stages. These constants are typically replaced by the corresponding constants for some reference distribution, or they are estimated.In this paper, it is argued that at least some stage of estimation for the constants is preferable to simply using the Normal reference without any estimation at all.Furthermore, it is seen that it is not enough to have only consistent estimates of the needed quantities.Sufficient rates of convergence for estimation of these constants to have negligible asymptotic effect are given.However, it is also noted that higher stages of estimation do not always give an improvement.