Distributional theory for the DIA method

The DIA method for the detection, identification and adaptation of model misspecifications combines estimation with testing. The aim of the present contribution is to introduce a unifying framework for the rigorous capture of this combination. By using a canonical model formulation and a partitioning of misclosure space, we show that the whole estimation-testing scheme can be captured in one single DIA estimator. We study the characteristics of this estimator and discuss some of its distributional properties. With the distribution of the DIA estimator provided, one can then study all the characteristics of the combined estimation and testing scheme, as well as analyse how they propagate into final outcomes. Examples are given, as well as a discussion on how the distributional properties compare with their usage in practice.