When constructing models in statistical estimation theory, we should bear in mind that a model is only an approximation of reality. It is therefore particularly interesting to consider stable models, i.e., models in which small changes of the “input data” lead to small changes “on the output”. There are at least two possible approaches to this topic. The first approach has been developed in robust estimation theory: it incorporates the disturbances directly in the construction of the model, i.e., the model is constructed so as to be stable under a particular class of disturbances. L. B. Klebanov [Principles of construction of models in parameter estimation theory. Ph. D. Diss., Leningrad (1986)] noted that we can describe the class of disturbances for which a given model is stable. He proposed a certain alternative approach relying on the choice of the loss function.
We investigate stability of the statistical model. Our approach is intermediate between the two previous approaches. We first choose a model (the method of estimation functions), and then construct a metric in the class of families of distributions such that small disturbances in this metric produce small disturbances in the asymptotic properties of the perturbed model.