Encompassing

The history of many sciences is characterized by an almost continuous emer­ gence of new theories. From a normative point of view, the survival of a new theory should mainly be determined by its ability to explain a new body of facts which the existing theories are unable to explain. If in addition the new theory is able to explain all the results obtained by the existing theories and if it can point out why these theories fail to explain certain facts, it should become the dominant theory. Otherwise, it might coexist with other theories for some time. Hence, a new theory ought to be judged not only by confronting it with existing facts, but also by confronting it with existing theories. The idea that a theory should be able to account for the results ob­ tained by other theories, although implicitly adhered to by many scientists, has rarely been formalized. The statistics literature on parametric hypoth­ esis testing, though, might be seen as an instance of such a formalization.

Contents: Pseudu-true values: Introduction Pseudo-true values.- Extension to conditional distributions.- Pseudo-true values of locations and scale parameters.- Examples.- Conclusion.- Encompassing: Introduction.- The data generating process and empirical models.- Definition of encompassing.- Properties of the encompassing relation.- Encompassing and parsimony as a model reduction device.- Examples.- Conclusion.- Testing the encompassing hypothesis: Introduction.- Encompassing and other statistical hypotheses.- The encompassing hypothesis in terms of almost sure limits.- Limit distributions of the basic statistics.- Limit distributions of quadratic forms.- Encompassing tests.- Nested models and implicit form formulation.- Conclusion.- Encompassing tests for the linear model: Introduction.- The framework.- Pseudo-true values and other limits.- Limit distributions.- Pseudo-ML estimators and basic statistics.- Encompassing tests.- Conclusion.