Equivalent kernels: One way of comparing linear smoothers like local-polynomial estimators and smoothing splines is to think of them as variants of the kernel estimator, where fitted values arise as...

Equivalent kernels: One way of comparing linear smoothers like local-polynomial estimators and smoothing splines is to think of them as variants of the kernel estimator, where fitted values arise as weighted averages of observed response values. This approach is illustrated in Figure 18.19, which shows equivalent kernel weights at two focal X-values in the Canadian occupational prestige data: One value, x(5), is near the boundary of the data; the other, x₍₆₀₎, is closer to the middle of the data. The figure shows tricube-kernel weights [panels (a) and (b)], along with the equivalent kernel weights for the local-linear estimator with span = 0:6 (or five equivalent parameters) [panels (c) and (d)] and the smoothing spline with five equivalent parameters [in panels (e) and (f)]. Compare and contrast the equivalent kernel weights for the three estimators. Are there any properties of the equivalent kernels for the local-linear and smoothing-spline estimators that you find surprising?