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# Testing the coherence of data and external intervals via an imprecise Sargan-Hansen test

*Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications*, PMLR 215:249-258, 2023.

#### Abstract

When information about a population is sparse, it is difficult to test whether a data set originated from that population. In applied research, however, researchers often have access to external information in the form of (central) statistical moments such as mean or variance. To compensate for the uncertainty in the external point values, this paper uses external intervals instead to represent the information about moments. The Sargan-Hansen test from the generalized method of moments framework is used, which exploits point-valued external information about moments in the presence of a statistical model to test whether data and external information are in conflict. For the Sargan-Hansen test, a separability result is derived with respect to the model and the external information. This result leads to a simplification of the test in terms of its analytical form and the calculation of the test statistics. To allow the use of external intervals instead of point values, an imprecise version of the Sargan-Hansen test is created using the Gamma-maximin decision rule. Assuming that the variables are normally distributed, a small sample version of this imprecise Sargan-Hansen test is derived. The power and type I errors of the developed tests are analyzed and compared in a simulation study in different small sample scenarios.