Penalized regression with model‐based penalties

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 t i, Y i is the minimizer of Σ{Yi ‐ μ( t i)} 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 ) = α o + α 1 t . Here the authors consider replacing ∫(μ″) 2 with the more general expression ∫( L μ) 2 where L is a linear differential operator with possibly nonconstant coefficients. The resulting estimate of μ performs well, particularly if L μ 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.