Variable Bandwidth Kernel Estimators of Regression Curves

In the model Yᵢ = g (tᵢ) + ᵢ, i = 1, , n, where Yᵢ are given observations, ᵢ i. i. d. noise variables and tᵢ nonrandom design points, kernel estimators for the regression function g (t) with variable bandwidth (smoothing parameter) depending on t are proposed. It is shown that in terms of asymptotic integrated mean squared error, kernel estimators with such a local bandwidth choice are superior to the ordinary kernel estimators with global bandwidth choice if optimal bandwidths are used. This superiority is maintained in a certain sense if optimal local bandwidths are estimated in a consistent manner from the data, which is proved by a tightness argument. The finite sample behavior of a specific local bandwidth selection procedure based on the Rice criterion for global bandwidth choice Rice (1984) is investigated by simulation.