## Penalized regression with model‐based penalties

### Abstract

Nonparametric regression techniques such as spline smoothing and local fitting depend implicitly on a parametric model. For instance, the cubic smoothing spline estimate of a regression function ∫ μ based on observations ti, Yi is the minimizer of Σ{Yi ‐ μ(ti)}2 + λ∫(μ′′)2. Since ∫(μ″)2 is zero when μ is a line, the cubic smoothing spline estimate favors the parametric model μ(t) = αo + α1t. Here the authors consider replacing ∫(μ″)2 with the more general expression ∫(Lμ)2 where L is a linear differential operator with possibly nonconstant coefficients. The resulting estimate of μ performs well, particularly if Lμ is small. They present an O(n) algorithm for the computation of μ. This algorithm is applicable to a wide class of L's. They also suggest a method for the estimation of L. They study their estimates via simulation and apply them to several data sets.

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