Open Access
2009 Jump estimation in inverse regression
Leif Boysen, Sophie Bruns, Axel Munk
Electron. J. Statist. 3: 1322-1359 (2009). DOI: 10.1214/08-EJS204


We consider estimation of a step function f from noisy observations of a deconvolution ϕ*f, where ϕ is some bounded L1-function. We use a penalized least squares estimator to reconstruct the signal f from the observations, with penalty equal to the number of jumps of the reconstruction. Asymptotically, it is possible to correctly estimate the number of jumps with probability one. Given that the number of jumps is correctly estimated, we show that for a bounded kernel ϕ the corresponding estimates of the jump locations and jump heights are n1/2 consistent and converge to a joint normal distribution with covariance structure depending on ϕ. As special case we obtain the asymptotic distribution of the least squares estimator in multiphase regression and generalizations thereof. Finally, singular integral kernels are briefly discussed and it is shown that the n1/2-rate can be improved.


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Leif Boysen. Sophie Bruns. Axel Munk. "Jump estimation in inverse regression." Electron. J. Statist. 3 1322 - 1359, 2009.


Published: 2009
First available in Project Euclid: 14 December 2009

zbMATH: 1326.62074
MathSciNet: MR2576315
Digital Object Identifier: 10.1214/08-EJS204

Primary: 62G05 , 62G20
Secondary: 42A82 , 46E22

Keywords: asymptotic normality , Change-point estimation , Deconvolution , entropy bounds , jump estimation , native Hilbert spaces , positive definite functions , reproducing kernel Hilbert spaces , singular kernels , total positivity

Rights: Copyright © 2009 The Institute of Mathematical Statistics and the Bernoulli Society

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