A Non-Parametric Cramér-von Mises Penalty Function Smoother

摘要

In many forms of penalty function smoothing, the choice of smoothing parameter is critical. Existing procedures for this choice include cross-validation or variations on likelihood methods. An alternative is introduced here, based on taking an intuitive non-parametric approach. For the linear spline form of penalty function, a Cramér-von Mises test statistic for the amount of smoothing is equal to the ratio of the least-squares and roughness components of the penalty formulation. Thus, constraining this test statistic to be equal to a central value of the Cramér-von Mises distribution provides a rational method for choosing the smoothing parameter. The necessary computations can be carried out by a convergent fixed-point iteration method, along with O(n) equation solving techniques facilitated by a Cholesky-like decomposition. An example is used to illustrate the method, and simulations show that it compares favorably to the use of cross-validation.