A note on generalized cross-validation with replicates

Generalized cross-validation (GCV) is a popular method for choosing the smoothing parameter in generalized spline smoothing when there are independent errors with common unknown variance. When data points are replicated, one can choose the smoothing parameter by minimizing one of three functions: the GCV score computed from the averaged observations, the GCV score computed from the original data, or an unbiased estimate of the risk using an independent estimate of the unknown variance [sigma]2. In this note we show how these three methods are related.