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Jump estimation in inverse regression. (English) Zbl 1326.62074

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.

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)
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