TY - JOUR

T1 - An improved Cp criterion for spline smoothing

AU - Chen, Chun Shu

AU - Huang, Hsin Cheng

PY - 2011/1

Y1 - 2011/1

N2 - Spline smoothing is a popular technique for curve fitting, in which selection of the smoothing parameter is crucial. Many methods such as Mallows' Cp, generalized maximum likelihood (GML), and the extended exponential (EE) criterion have been proposed to select this parameter. Although Cp is shown to be asymptotically optimal, it is usually outperformed by other selection criteria for small to moderate sample sizes due to its high variability. On the other hand, GML and EE are more stable than Cp, but they do not possess the same asymptotic optimality as Cp. Instead of selecting this smoothing parameter directly using Cp, we propose to select among a small class of selection criteria based on Stein's unbiased risk estimate (SURE). Due to the selection effect, the spline estimate obtained from a criterion in this class is nonlinear. Thus, the effective degrees of freedom in SURE contains an adjustment term in addition to the trace of the smoothing matrix, which cannot be ignored in small to moderate sample sizes. The resulting criterion, which we call adaptive Cp, is shown to have an analytic expression, and hence can be efficiently computed. Moreover, adaptive Cp is not only demonstrated to be superior and more stable than commonly used selection criteria in a simulation study, but also shown to possess the same asymptotic optimality as Cp.

AB - Spline smoothing is a popular technique for curve fitting, in which selection of the smoothing parameter is crucial. Many methods such as Mallows' Cp, generalized maximum likelihood (GML), and the extended exponential (EE) criterion have been proposed to select this parameter. Although Cp is shown to be asymptotically optimal, it is usually outperformed by other selection criteria for small to moderate sample sizes due to its high variability. On the other hand, GML and EE are more stable than Cp, but they do not possess the same asymptotic optimality as Cp. Instead of selecting this smoothing parameter directly using Cp, we propose to select among a small class of selection criteria based on Stein's unbiased risk estimate (SURE). Due to the selection effect, the spline estimate obtained from a criterion in this class is nonlinear. Thus, the effective degrees of freedom in SURE contains an adjustment term in addition to the trace of the smoothing matrix, which cannot be ignored in small to moderate sample sizes. The resulting criterion, which we call adaptive Cp, is shown to have an analytic expression, and hence can be efficiently computed. Moreover, adaptive Cp is not only demonstrated to be superior and more stable than commonly used selection criteria in a simulation study, but also shown to possess the same asymptotic optimality as Cp.

KW - Effective degrees of freedom

KW - Selection effect

KW - Smoothing spline

KW - Stein's unbiased risk estimate

UR - http://www.scopus.com/inward/record.url?scp=77956281310&partnerID=8YFLogxK

U2 - 10.1016/j.jspi.2010.06.017

DO - 10.1016/j.jspi.2010.06.017

M3 - 期刊論文

AN - SCOPUS:77956281310

SN - 0378-3758

VL - 141

SP - 445

EP - 452

JO - Journal of Statistical Planning and Inference

JF - Journal of Statistical Planning and Inference

IS - 1

ER -