Electronic Journal of Statistics

Jump estimation in inverse regression

Leif Boysen, Sophie Bruns, and Axel Munk

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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.

Article information

Electron. J. Statist., Volume 3 (2009), 1322-1359.

First available in Project Euclid: 14 December 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G20: Asymptotic properties
Secondary: 42A82: Positive definite functions 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]

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


Boysen, Leif; Bruns, Sophie; Munk, Axel. Jump estimation in inverse regression. Electron. J. Statist. 3 (2009), 1322--1359. doi:10.1214/08-EJS204. https://projecteuclid.org/euclid.ejs/1260801226

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