Testing the fit of data and external sets via an imprecise Sargan-Hansen test

In empirical sciences such as psychology, the term cumulative science mostly refers to the integration of theories, while external (prior) information may also be used in statistical inference. This external information can be in the form of statistical moments and is subject to various types of uncertainty, e.g., because it is estimated, or because of qualitative uncertainty due to differences in study design or sampling. Before using it in statistical inference, it is therefore important to test whether the external information fits a new data set, taking into account its uncertainties. As a frequentist approach, the Sargan-Hansen test from the generalized method of moments framework is used in this paper. It tests, given a statistical model, whether data and point-wise external information are in conflict. A separability result is given that simplifies the Sargan-Hansen test statistic in most cases. The Sargan-Hansen test is then extended to the imprecise scenario with (estimated) external sets using stochastically ordered credal sets. Furthermore, an exact small sample version is derived for normally distributed variables. As a Bayesian approach, two prior-data conflict criteria are discussed as a test for the fit of external information to the data. Two simulation studies are performed to test and compare the power and type I error of the methods discussed. Different small sample scenarios are implemented, varying the moments used, the level of significance, and other aspects. The results show that both the Sargan-Hansen test and the Bayesian criteria control type I errors while having sufficient or even good power. To facilitate the use of the methods by applied scientists, easy-to-use R functions are provided in the R script in the supplementary materials.