Summary: Nonparametric regression, often called smoothing, is a widely used data analysis method. The use of a smoother requires the choice of a smoothing parameter that by balancing fidelity and roughness controls how much smoothing is done. Two popular selection criteria for choosing the smoothing parameter are \(C_p\) and generalized maximum likelihood (GML). Each of these has its own problems. For \(C_p\), the problem is its high variability, whereas for GML, the problem is its potentially large bias.
By studying the geometry of selection criteria, we give an intuitive explanation of the strength and weakness of \(C_p\) and GML. The geometry then motivates a new selection method, the extended exponential (EE) criterion, which combines the strength of \(C_p\) and GML but mitigates their weaknesses in terms of variability, bias, and undersmoothing.