Nonparametric Curve Estimation by Smoothing Splines: Unbiased-Risk-Estimate Selector and its Robust Version via Randomized Choices
Résumé
This Demonstration considers a simple nonparametric regression problem: how to recover a function f of one variable, here over [0,1], when only n couples (Subscript[x, i],Subscript[y, i]) are known for i==1,2,\[Ellipsis] ,n that satisfy the model Subscript[y, i]=f(Subscript[x, i])+\[Sigma] Subscript[\[Epsilon], i], where Subscript[x, i]\[Element][0,1] and the Subscript[\[Epsilon], i] are independent, standard normal random variables. For simplicity, assume that the variance \[Sigma]^2 is also known.
In this Demonstration, we have implemented the randomization-based method introduced by the author and analyzed in [D. A. Girard, "Estimating the Accuracy of (Local) Cross-Validation via Randomised GCV Choices in Kernel or Smoothing Spline Regression," Journal of Nonparametric Statistics, 22(1), 2010 pp. 41-64. doi:10.1080/10485250903095820, section 7.2], which permits computing a "more parsimonious yet 'near-optimal' fit". Such a fit is parameterized by a percentile p>50, which determines an upward modification of the original Subscript[C, L] choice.