Optimal Fixing of the Bandwidth‐Parameter for the Empirical Regression1,2

Abstract

AbstractThe estimator ř0(x) of the regression r(x) = E (Y | × = x) from measured points (xi, yi), i = 1(1) n, of a continuous two‐dimensional random variable (X, Y) with unknown continuous density function f(x, y) and with moments up to the second order can be made with the help of a density estimation f̌0(x, y) (see e.g. SCHMERLING and PEIL, 1980). Here f̌0(x, y) still contains free parameters (so‐called band‐width‐parameters), the values of which have to be optimally fixed in the concrete case. This fixing can be done by using a modification of the maximum‐likelihood principle including jackknife techniques. The parameter values can be also found from the estimators for r(x). Here the cross‐validation principle can be applied. Some numerical aspects of these possibilities for optimally fixing the bandwidth‐parameter are discussed by means of examples. If ř0(x) is used as a smoothing operator for time series the optimal choice of the parameter values is dependent on the purpose of application of the smoothed time series. The fixing will then be done by considering the so‐called filter‐characteristic of řC0(x).