A comparative study of several smoothing methods in density estimation

A simple root n bandwidth selector for nonparametric regression

“…For n = 50 hupi and h<DS\ perform quite similar. For n = 100 hpsi is better than h^pi for all three regressions, following the criterion of Averaged Squared Error to h 0 , ASE(h 0 ) =: j. J2j=i(hj ~ ho) 2 -Now both, the bias and the standard deviation of hus\ are smaller than the ones of hupi-This conforms with the theoretical results, because the rate of convergence of hpsi is higher. We think the difference will be more evident if a simulation with larger n is done.…”

“…The proposal in HHM is a double-smoothing (DS) procedure. The consideration of this method is similar to those of the smoothed croos-validation and the smoothed bootstrap, both of them show a fairly satisfactory performance in the context of kernel density estimation (Cao, Cuevas and Gonzalez-Manteiga, 1994). Under certain conditions the bandwidth selectors of HHM are root n consistent.…”

“…Assumption 3. a 2 is root n consistent for <r 2 , that is, The optimal choices of v and 8 induce a linear constraint between them so if one of them is given the other can easily be obtained. In the following we discuss some special cases.…”

“…There is a variance term, n~3a 2 Y21 E"=i Aj(x{) 2 , in this estimation, which was subtracted in HHM. This term does not depend on the data and plays a similar role as the nonstochastic term in Jones and Sheather (1991).…”

“…The second generation methods, including various new plug-in methods (e.g. Marron, 1990 and, smoothed cross-validation (Hall, Marron and Park, 1992), smoothed bootstrap (Marron, 1992, Cao, 1993and Cao, Cuevas and Gonzalez-Manteiga, 1994 and some root n convergent methods (Jones, Marron and Park, 1991 (JMP) and Marron, 1991), are far superior to the better known first generation methods.…”