Nonparametric Estimation with Local Rules

An attempt was made to identify the questions which are of genuine importance to a statistician who is interested in evaluating nonparametric estimation rules. The asymptotic performance of various nearest neighbor rules and loss functions was analyzed, and the results concerning the convergence of the conditional risk are quite strong considering the simple nature of the rules and the minimal assumptions made concerning the problem structure. In addition, distribution-free bounds on the error of two different estimates of finite sample performance are derived for a class of estimation rules which include k-nearest neighbor rules. For comparison purposes, a simulation study was carried out so that the bounds for a few specific distributions could be compared with the theoretical bounds.