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Simultaneous confidences bands with the volume-of-tube formula and spline estimators

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Simultaneous confidence bands for a smooth curve are typically based on limit theorems for the supremum of standardized deviation of the true function from its nonparametric estimator and related Gaussian processes. However, the convergence of suprema of Gaussian processes to the limiting extreme value distribution is exceedingly slow, so that in finite samples some bootstrap approximations need to be used, which is rather computationally intensive. Moreover, in practice the unknown bias and the large variability of an estimated smoothing parameter complicate the matter further. Alternatively, one can consider a two-term approximation to the tail probability of suprema of Gaussian processes (Sun, 1993), which turned out to be connected to the volume of a tube around a manifold embedded in a unit sphere. This approximation is much more accurate and requires no bootstrap even in small samples. In this talk we consider simultaneous confidence bands based on the volume-of-tube formula for a smooth curve estimated with penalized splines. In particular, we discuss how the unknown bias and the variability of an estimated smoothing parameter can be handled using the (empirical) Bayesian formulation of spline estimators. This is the joint work with Thomas Kneib and Gerda Claeskens.

http://www.uni-goettingen.de/de/101995.html

This talk is part of the Statistics series.

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