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Testing lack-of-fit in inverse regression models

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We propose two test statistics for use in inverse regression problems where only noisy, indirect observations for the mean function are available. Both test statistics have a counterpart in classical hypothesis testing, where they are called the order selection test and the data-driven Neyman smooth test. We also introduce two model selection criteria which extend the classical AIC and BIC to inverse regression problems. In a simulation study we show that the inverse order selection and Neyman smooth tests outperform their direct counterparts in many cases. The methods are applied to data arising in confocal fluorescence microscopy. Here, images are observed with blurring (modeled as deconvolution) and stochastic error at subsequent times. The aim is then to reduce the signal to noise ratio by averaging over the distinct images. In this context it is relevant to test whether the images are still equal (or have changed by outside influences such as moving of the object table). This is joint work with N. Bissantz, H. Holzmann and A. Munk.

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