# Empirical Bayesian test of the smoothness

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- Eurandom
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## Abstract

EMPIRICAL BAYESIAN TEST OF THE SMOOTHNESS By Eduard Belitser and Farida Enikeeva Utrecht University and EURANDOM 7th April 2005 In the context of adaptive nonparametric curve estimation problem, a common as- sumption is that a function (signal) to estimate belongs to a nested family of func- tional classes, parameterized by a quantity which often has a meaning of smoothness amount. It has already been realized by many that the problem of estimating the smoothness is not sensible. What then can be inferred about the smoothness? The paper attempts to answer this question. We consider the implications of our results to hypothesis testing. We also relate them to the problem of adaptive estimation. The test statistic is based on the marginalized maximum likelihood estimator of the smoothness for an appropriate prior distribution on the unknown signal. 1 Introduction Suppose we observe independent Gaussian data X = (Xi)i∈N, where Xi ∼ N (θi, n−1), θ = (θi)i∈N ∈ `2 is an unknown parameter. This model is the sequence version of the Gaussian white noise model dY (t) = f(t)dt+n−1/2dW (t), t ∈ [0, 1], where f ∈ L2[0, 1] = L2 is an unknown signal and W is the standard Brownian motion. The infinite dimensional parameter θ ∈ `2 can be regarded as the sequence of the Fourier coefficients of f ∈ L2 with respect to some orthonormal basis in L2. Sometimes we will call θ a signal. The white noise model has received much attention in the last few decades and compre- hensive treatments of it can be found in Ibragimov and Khasminski (1981) and Johnstone (1999). Besides of being of interest on its own (the problem of recovering a signal transmitted over a communication channel with Gaussian white noise of intensity n−1/2), the white noise model turns out to be a mathematical idealization of some other nonparametric models. For instance, the white noise model arises as the limiting experiment as n → ∞, for the model of n i.i.d. observations with unknown density (see Nussbaum (1996), Grama and Nussbaum

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