Variable Bandwidth Kernel Estimators of Regression Curves

In the model Yᵢ = g(tᵢ) + εᵢ, i = 1,⋯, n, where Yᵢ are given observations, εᵢ 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.