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...
As journalists are expected to report on events where expectations and rules are transgressed, they often report on moral violations (such as murder, tax evasion, or unjust political decisions). Exposed to journalistic reports on violations of their moral principles, individuals instantly feel that these actions are wrong. According to theories of morality, immorality perceptions are associated with specific cognitive and affective reactions. In two studies, we used the concept of a moral dyad to (a) define moral news content and (b) analyze emotional reactions and memory effects of intuitive perceptions of immorality. In both studies, immorality led to higher levels of anger and compassion, but impaired memory with effects hinging on perception of immorality. These perceptions further did not differ across different presentations of dyads. Our findings show the usefulness to employ a lens of morality to look at the entire news production and reception process.
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