Testing for lack of fit in inverse regression-with applications to biophotonic imaging

• 成果类型: Article
• 署名作者: Bissantz, Nicolai; Claeskens, Gerda; Holzmann, Hajo; Munk, Axel
• 署名单位: Ruhr University Bochum; KU Leuven; Helmholtz Association; Karlsruhe Institute of Technology; University of Gottingen
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• DOI: 10.1111/j.1467-9868.2008.00670.x
• 发表日期: 2009
• 页码: 25-48
• 关键词: GOODNESS-OF-FIT; data-driven version; smooth test; deconvolution; CONVERGENCE; density; rates

摘要

We propose two test statistics for use in inverse regression problems Y=K theta+epsilon, where K is a given linear operator which cannot be continuously inverted. Thus, only noisy, indirect observations Y for the function theta 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.