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October, 1995 Variational Solution of Penalized Likelihood Problems and Smooth Curve Estimation
Martin B. Machler
Ann. Statist. 23(5): 1496-1517 (October, 1995). DOI: 10.1214/aos/1176324309


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.


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Martin B. Machler. "Variational Solution of Penalized Likelihood Problems and Smooth Curve Estimation." Ann. Statist. 23 (5) 1496 - 1517, October, 1995.


Published: October, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0843.62044
MathSciNet: MR1370293
Digital Object Identifier: 10.1214/aos/1176324309

Primary: 62G07
Secondary: 34B10 , 41A29 , 65D07 , 65D10

Keywords: inflection point , maximum penalized likelihood , Nonparametric regression , robust smoothing , roughness penalty

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 5 • October, 1995
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