Electronic Journal of Statistics
- Electron. J. Statist.
- Volume 3 (2009), 1322-1359.
Jump estimation in inverse regression
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 n−1/2 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 n−1/2-rate can be improved.
Electron. J. Statist., Volume 3 (2009), 1322-1359.
First available in Project Euclid: 14 December 2009
Permanent link to this document
Digital Object Identifier
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