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Testing for lack of fit in inverse regression—with applications to biophotonic imaging
Author(s) -
Bissantz Nicolai,
Claeskens Gerda,
Holzmann Hajo,
Munk Axel
Publication year - 2009
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/j.1467-9868.2008.00670.x
Subject(s) - akaike information criterion , mathematics , context (archaeology) , bayesian information criterion , inverse problem , statistics , inverse , model selection , bayes' theorem , regression , computer science , algorithm , artificial intelligence , bayesian probability , mathematical analysis , paleontology , geometry , biology
Summary. We propose two test statistics for use in inverse regression problems Y = K θ + ɛ , where K is a given linear operator which cannot be continuously inverted. Thus, only noisy, indirect observations Y for the function θ 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 Akaike information criterion and Bayes information criterion 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, modelled as convolution, 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.