The exponentially tilted family of densities is used to discuss the relationship between the encompassing principle and the <italic>M</italic>-test or conditional moment testing principle. It is shown that the two principles are capable of generating the same test statistics and in this sense equivalent....

A model M is said to encompass another model N if the former can explain the results obtained by the latter. In this paper, we propose a general notion of encompassing that covers both classical and Bayesian viewpoints and essentially represents a concept of sufficiency among models. We...