Abstract
Usual nonparametric regression estimators often show many little wiggles which do not appear to be necessary for a good description of the data. The new "Wp" smoother is a maximum penalized likelihood (MPL) estimate with a novel roughness penalty. It penalizes a relative change of curvature. This leads to disjoint classes of functions, each with given number, $n_w$, of inflection points. For a "Wp" estimate, $f"(x) = \pm (x - w_1)\cdots (x - w_{n_w}) \cdot \exp h_f(x)$, which is semiparametric, with parameters $w_j$ and nonparametric part $h_f(\cdot)$. The main mathematical result is a convenient form of the characterizing differential equation for a very general class of MPL estimators.
Citation
Martin B. Machler. "Variational Solution of Penalized Likelihood Problems and Smooth Curve Estimation." Ann. Statist. 23 (5) 1496 - 1517, October, 1995. https://doi.org/10.1214/aos/1176324309
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