Testing for Lack of Fit in Inverse Regression - With Applications to Photonic Imaging

We propose two test statistics for use in inverse regression problems Y = Kcedil; + , where K is a given matrix or operator which cannot be continuously inverted. Thus, only noisy, indirect observations Y for the function cedil; are available. The tests are designed for hypotheses of the form H0 : cedil;(x) = p j=1 ajAElig;j (x), where (AElig;j )je1 is the orthonormal system of basis functions given by the spectral decomposition of the operator K. 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 theory is motivated by 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 decide whether the images are still equal (or have changed by outside influences such as moving of the object table)