A Comparison of GCV and GML for Choosing the Smoothing Parameter in the Generalized Spline Smoothing Problem

摘要: The partially improper prior behind the smoothing spline model is used to obtain a generalization of maximum likelihood (GML) estimate for parameter. Then this compared with generalized cross validation (GCV) both analytically and by Monte Carlo methods. comparison based on predictive mean square error criteria. It shown that if true, unknown function being estimated smooth in sense be defined then GML undersmooths relative GCV using goes zero at slower rate than estimate. If true "rough" estimates have asymptotically similar behavior. A experiment was designed see asymptotic results case were evident small sample sizes. Mixed obtained $n = 32$, somewhat better 64$, decidedly superior 128$. In 32$ smaller $\sigma^2$ close larger $\sigma^2$. theoretical are extend model, which includes functions given noisy values various integrals them.