Estimating the mean function of a Gaussian process and the Stein effect
By James O. Berger, Robert L. Wolpert. 1983
The problem of global estimation of the mean function [theta](·) of a quite arbitrary Gaussian process is considered. The loss function in estimating [theta] by a function a(·) is assumed to be of the form L([theta], a) = [integral operator] [[theta](t) - a(t)]2[mu](dt), and estimators are evaluated in terms of their risk function (expected loss). The usual minimax estimator of [theta] is shown to be inadmissible via the Stein phenomenon; in estimating the function [theta] we are trying to simultaneously estimate a larger number of normal means. Estimators improving upon the usual minimax estimator are constructed, including an estimator which allows the incorporation of prior information about [theta]. The analysis is carried out by using a version of the Karhunen-Loeve expansion to represent the original problem as the problem of estimating a countably infinite sequence of means from independent normal distributions.
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- Anthony R'eveillac
- Nicolas Privault
- Ivan Nourdin
- Giovanni Peccati
- Günter Last
- Ilya S. Molchanov
- Matthias Schulte
Citations in the corpus
- Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “A direct generalization of the James-Stein estimator to an infinite dimensional Hilbert space has been considered in Berger and Wolpert (1983), Privault and Réveillac (2008), Mandelbaum and Shepp (1987).”
- Kernel Mean Shrinkage Estimators (2014, methodology): “A direct generalization of James-Stein estimator to infinite-dimensional Hilbert space has been considered, for example, in Berger and Wolpert (1983); Mandelbaum and Shepp (1987); Privault and Rveillac (2008).”
- Kernel Mean Estimation and Stein's Effect (2013, neutral): “Although some attempts have been made to understand this phenomenon in the infinite-dimensional space [17, 18, 19], this work presents a key challenge: the true probability distribution of μP is essentially unknown and is subject to the choice of kernel k and distribution P, whereas all the previous”
- Stein estimation for the drift of Gaussian processes using the Malliavin calculus (2008, background): “1 where the drift (ut)t∈[0,T ] is viewed as an input signal to be estimated and perturbed by a centered Gaussian noise (Xut )t∈[0,T ]; see, for example, [4], Chapter VII.”