In this chapter, we return to smoothing splines of arbitrary order but now for general, nonparametric regression problems with random designs. One goal is indeed to rework parts of Chapter 13, but we implement it as a byproduct of a more ambitious project regarding “asymptotically equivalent” (or just “equivalent”) kernel approximations to smoothing splines. We interpret “equivalence” in the strict sense that the (global) bias and variance of the difference between the equivalent kernel and smoothing spline estimators are asymptotically negligible compared with the bias and variance of either estimator. (It would seem that under any weaker interpretation all “good” nonparametric regression estimators would be equivalent to one another.) An interesting twist in the “equivalent” kernel story is that the bias and the noise components require different representations for the “equivalence” to hold in the strict sense.
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© 2009 Springer-Verlag New York
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Eggermont, P.P.B., LaRiccia, V.N. (2009). Equivalent Kernels for Smoothing Splines. In: Maximum Penalized Likelihood Estimation. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/b12285_10
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DOI: https://doi.org/10.1007/b12285_10
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