Shape constrained estimators in inverse regression models with convolution type operator

Abstract

In this paper we are concerned with shape restricted estimation in inverse regression problems with convolution-type operator. We use increasing rearrangements to compute increasing and convex estimates from an (in principle arbitrary) unconstrained estimate of the unknown regression function. An advantage of our approach is that it is not necessary that prior shape information is known to be valid on the complete domain of the regression function. Instead, it is sufficient if it holds on some compact interval. A simulation study shows that the shape restricted estimate on the respective interval is significantly less sensitive to moderate undersmoothing than the unconstrained estimate, which substantially improves applicability of estimates based on data-driven bandwidth estimators. Finally, we demonstrate the application of the increasing estimator by the estimation of the luminosity profile of an elliptical galaxy. Here, a major interest is in reconstructing the central peak of the profile, which, due to its small size, requires to select the bandwidth as small as possible.