A Stochastic Estimator of the Trace of the Influence Matrix for Laplacian Smoothing Splines

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Abstract
An unbiased stochastic estimator of tr(I–A), where A is the influence matrix associated with the calculation of Laplacian smoothing splines, is described. The estimator is similar to one recently developed by Girard but satisfies a minimum variance criterion and does not require the simulation of a standard normal variable. It uses instead simulations of the discrete random variable which takes the values 1, -1 each with probability 1/2. Bounds on the variance of the estimator, similar to those established by Girard, are obtained using elementary methods. The estimator can be used to approximately minimize generalised cross validation (GCV) when using discretized iterative methods for fitting Laplacian smoothing splines to very large data sets. Simulated examples show that the estimated trace values, using either the estimator presented here or the estimator of Girard, perform almost as well as the exact values when applied to the minimization of GCV for n as small as a few hundred, where n is the number of data points.