We study the asymptotics for jump-penalized least squares regression aiming at approximating a regression function by piecewise constant functions. Besides conventional consistency and convergence rates of the estimates in L 2 ([0, 1)) our results cover other metrics like Skorokhod metric on the space of càdlàg functions and uniform metrics on C([0, 1]). We will show that these estimators are in an adaptive sense rate optimal over certain classes of "approximation spaces." Special cases are the class of functions of bounded variation (piecewise) Hölder continuous functions of order 0 < α ≤ 1 and the class of step functions with a finite but arbitrary number of jumps. In the latter setting, we will also deduce the rates known from change-point analysis for detecting the jumps. Finally, the issue of fully automatic selection of the smoothing parameter is addressed.

We consider estimation of a step function f from noisy observations of a deconvolution φ * f , where φ 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 the corresponding parameter 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, and that this rate is minimax for bounded continuous kernels φ. As special case we obtain the asymptotic distribution of the least squares estimator in multiphase regression and generalisations thereof. In contrast to the results obtained for bounded φ, we show that for kernels with a singularity of order O(|x| −α ), 1/2 < α < 1, a jump location can be estimated at a rate of n −1/(3−2α) , which is again the minimax rate. We find that these rate do not depend on the spectral information of the operator rather on its localization properties in the time domain. Finally, it turns out that adaptive sampling does not improve the rate of convergence, in strict contrast to the case of direct regression. AMS 2000 subject classifications: Primary 62G05, 62G20; secondary 42A82, 46E22. 1 imsart-generic ver. 2008/01/24 file: ejs_2008_204.tex date: November 21, 2018 L. Boysen and A. Munk/Jumps in inverse regression 2 reproducing kernel Hilbert spaces, minimax rates, adaptive sampling, optimal design. imsart-generic ver. 2008/01/24 file: ejs_2008_204.tex date: November 21, 2018 L. Boysen and A. Munk/Jumps in inverse regression 4 Butucea and Tsybakov (2007)). However, we stress that in many practical situations, gaussian deconvolution is still applied, leading to satisfactory results (see e.g. Bissantz et al. (2007) for an example in astrophysics). At a first glance this seems to be contradictory. However, often a minimax result leads to rather pessimistic view, in particular in large function classes such as Sobolev spaces are.Often, more restrictive modeling is possible and necessary to obtain reasonably good rates of convergence. In fact the space of locally constant functions as considered in this paper (albeit of dimension ∞) yields a n −1/2 rate of convergence generically which renders deconvolution in this setting as a practically feasable task. In fact, in this case the correct (and finite) number of jumps will be estimated asymptotically, and the problem reduces to a (nonsmooth) nonlinear regression problem.We will give general conditions, which are sufficient to deduce the n −1/2 rate.These conditions are borrowed from the theory of radial basis functions in native Hilbert spaces and from total positivity. They cover super-smooth functions such as the Gauss-kernel, polynomial kernels φ(x) = x p 1 [0,1) (x) with p = 0, 1, . . . and continuous symmetric functions φ which have a Fourier transfo...

We study the asymptotic behavior of piecewise constant least squares regression estimates, when the number of partitions of the estimate is penalized. We show that the estimator is consistent in the relevant metric if the signal is in L 2 ([0, 1]), the space of càdlàg functions equipped with the Skorokhod metric or C([0, 1]) equipped with the supremum metric. Moreover, we consider the family of estimates under a varying smoothing parameter, also called scale space. We prove convergence of the empirical scale space towards its deterministic target.

We derive the asymptotic distribution of the integrated square error of a deconvolution kernel density estimator in supersmooth deconvolution problems. Surprisingly, in contrast to direct density estimation as well as ordinary smooth deconvolution density estimation, the asymptotic distribution is no longer a normal distribution but is given by a normalized chi-squared distribution with 2 d.f. A simulation study shows that the speed of convergence to the asymptotic law is reasonably fast. Copyright 2006 Board of the Foundation of the Scandinavian Journal of Statistics..

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