| 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 integral mu based on observations t(i), Y-i is the minimizer of Sigma {Y-i - mu>(*) over bar * (t(i))}(2) + lambda integral>(*) over bar *(mu'')(2). Since integral>(*) over bar *(mu'')(2) is zero when mu is a line, the cubic smoothing spline estimate favors the parametric model mu>(*) over bar * (t) = alpha (0) + alpha (1)t. Here the authors consider replacing integral>(*) over bar *(mu'')(2) with the mon general expression integral>(*) over bar * (L mu)(2) where L is a linear differential operator with possibly nonconstant coefficients. The resulting estimate of mu performs well, particularly if L mu is small. They present an O(n) algorithm far the computation of mu. 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. |
