Statistical inference for the optimal approximating model

Rohde, Angelika; Dümbgen, Lutz

doi:10.1007/s00440-012-0414-7

Cited by 4 publications

(6 citation statements)

References 25 publications

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“…χ 2 1 random variables. By Proposition 6 in Rohde and Dümbgen [21] (similar statements have been derived also elsewhere, for another reference see, for instance, Johnstone [17], p. 74), for a vector (µ i ) i∈An of real-valued numbers…”

Section: Appendix: Proofssupporting

confidence: 64%

Asymptotically efficient estimation of a scale parameter in Gaussian time series and closed-form expressions for the Fisher information

Sabel¹,

Schmidt-Hieber²

2014

Bernoulli

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Mimicking the maximum likelihood estimator, we construct first order Cramer-Rao efficient and explicitly computable estimators for the scale parameter σ 2 in the model Zi,n = σn −β Xi + Yi, i = 1, . . . , n, β > 0 with independent, stationary Gaussian processes (Xi) i∈N , (Yi) i∈N , and (Xi) i∈N exhibits possibly long-range dependence. In a second part, closed-form expressions for the asymptotic behavior of the corresponding Fisher information are derived. Our main finding is that depending on the behavior of the spectral densities at zero, the Fisher information has asymptotically two different scaling regimes, which are separated by a sharp phase transition. The most prominent example included in our analysis is the Fisher information for the scaling factor of a high-frequency sample of fractional Brownian motion under additive noise.

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Section: Appendix: Proofssupporting

confidence: 64%

Asymptotically efficient estimation of a scale parameter in Gaussian time series and closed-form expressions for the Fisher information

Sabel¹,

Schmidt-Hieber²

2014

Bernoulli

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“…Combining the above with the exponential inequality for χ 2 -squared random variables found in Proposition 6 of [46], we have…”

Section: Results For 2 -Settingmentioning

confidence: 57%

Adaptive Bernstein–von Mises theorems in Gaussian white noise

Ray¹

2017

Ann. Statist.

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We investigate Bernstein-von Mises theorems for adaptive nonparametric Bayesian procedures in the canonical Gaussian white noise model. We consider both a Hilbert space and multiscale setting with applications in L 2 and L ∞ respectively. This provides a theoretical justification for plug-in procedures, for example the use of certain credible sets for sufficiently smooth linear functionals. We use this general approach to construct optimal frequentist confidence sets based on the posterior distribution. We also provide simulations to numerically illustrate our approach and obtain a visual representation of the geometries involved.

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“…In the following we will briefly discuss other deviation inequalities for χ 2 distributed random variables [7,8,10,27,42,49] and their connections to (36) from the perspective of our purpose. At first we mention an inequality by Birgé [8,Lemma 8.1], which deals with k = 1 and coincides with the above result in that case.…”

Section: A a Chi-squared Deviation Inequalitymentioning

confidence: 99%

“…for any ξ > 0. Then the weak law of large numbers for triangular arrays (the condition (42) implies the conditions imposed in [44, Ex. 2.2]) is applicable (as |I n | 0 and hence l n ∞), i.e.…”

Section: B Proofs Of Sectionmentioning

confidence: 99%

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Bump detection in heterogeneous Gaussian regression

Enikeeva¹,

Munk²,

Werner³

2018

Bernoulli

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We analyze the effect of a heterogeneous variance on bump detection in a Gaussian regression model. To this end we allow for a simultaneous bump in the variance and specify its impact on the difficulty to detect the null signal against a single bump with known signal strength. This is done by calculating lower and upper bounds, both based on the likelihood ratio.Lower and upper bounds together lead to explicit characterizations of the detection boundary in several subregimes depending on the asymptotic behavior of the bump heights in mean and variance. In particular, we explicitly identify those regimes, where the additional information about a simultaneous bump in variance eases the detection problem for the signal. This effect is made explicit in the constant and / or the rate, appearing in the detection boundary.We also discuss the case of an unknown bump height and provide an adaptive test and some upper bounds in that case.

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Statistical inference for the optimal approximating model

Cited by 4 publications

References 25 publications

Asymptotically efficient estimation of a scale parameter in Gaussian time series and closed-form expressions for the Fisher information

Asymptotically efficient estimation of a scale parameter in Gaussian time series and closed-form expressions for the Fisher information

Adaptive Bernstein–von Mises theorems in Gaussian white noise

Bump detection in heterogeneous Gaussian regression

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