One approach to the construction of stable estimation procedures (Q803687)

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. \textit{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.