Abstract

The application of characterization theorems for construction of goodnessof-fit tests dates to the middle of the last century and recently has become one of the main directions in this field. The growth of the number of characterization theorems significantly contributed to the development of this direction. Such tests are attractive because they reflect some intrinsic properties of probability distributions connected with the given characterization, and therefore can be more efficient or more robust than others. Several approaches for the construction of such tests, based on different types of characterizations, such as equidistribution-type, independence-type,momenttype, functional equation-type, will be presented. All presented general results will be accompanied by concrete examples of exponentially and symmetry tests, and their possible applications in time series analysis. It would be also shown how the constructed test can be compared in terms of Bahadur asymptotic efficiency. Finally, some possible directions for future research will be shown