Bandwidth Selection: Classical or Plug-In?

Bandwidth selection for procedures such as kernel density estimation and local regression have been widely studied over the past decade. Substantial "evidence" has been collected to establish superior performance of modern plug-in methods in comparison to methods such as cross validation; this has ranged from detailed analysis of rates of convergence, to simulations, to superior performance on real datasets. In this work we take a detailed look at some of this evidence, looking into the sources of differences. Our findings challenge the claimed superiority of plug-in methods on several fronts. First, plug-in methods are heavily dependent on arbitrary specification of pilot bandwidths and fail when this specification is wrong. Second, the often-quoted variability and undersmoothing of cross validation simply reflects the uncertainty of bandwidth selection; plug-in methods reflect this uncertainty by oversmoothing and missing important features when given difficult problems. Third, we look at asymptotic theory. Plug-in methods use available curvature information in an inefficient manner, resulting in inefficient estimates. Previous comparisons with classical approaches penalized the classical approaches for this inefficiency. Asymptotically, the plug-in based estimates are beaten by their own pilot estimates.