Boysen, Leif; Bruns, Sophie; Munk, Axel Jump estimation in inverse regression. (English) Zbl 1326.62074 Electron. J. Stat. 3, 1322-1359 (2009). Summary: We consider estimation of a step function \(f\) from noisy observations of a deconvolution \(\phi \ast f\), where \(\phi\) is some bounded \(L_{1}\)-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 \(\phi \) 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 \(\varphi\). 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. Cited in 3 Documents MSC: 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference 46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) Keywords:change-point estimation; deconvolution; jump estimation; asymptotic normality; positive definite functions; native Hilbert spaces; entropy bounds; reproducing kernel Hilbert spaces; singular kernels; total positivity × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: [1] Achieser, N. I. (1992)., Theory of approximation . Dover Publications Inc., New York. Translated from the Russian and with a preface by Charles J. Hyman, Reprint of the 1956 English translation. · Zbl 0072.28403 [2] Birgé, L. and Massart, P. (2007). Minimal penalties for gaussian model selection., Probab. 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