Divergence and model adequacy, a semiparametric case study

• LPSM, Sorbonne University, Paris, France [michel.broniatowski@sorbonne-universite.fr]

• Paris []

Keywords: Guidelines – ICORS – Robustness

1 Abstract

Choosing a statistical inferetial criterion remains a complex question when addressing non regular or non/semi parametric models. Depending on the standpoint underlying the statistical analysis, the choice may result in robust properties, accuracy or more often on numerical faisibility. In the present talk we focus on adequacy between the criterion at hand and the analytic properties of the model, in some semi parametric context. As such this is a case study, whose aim is to propose a few guidelines towards adequacy between statistical criterion and model when defined through characterizations in terms of regularity of the distributions.

The situation gets interesting and complex when the model is a collection of subsets of the class of all distributions with non void interior, and this collection is indexed by a finite dimensional parameter, which is the parameter of interest; a simple example is when each of these subsets consists in all distributions with same expectation, which is the current value of the parameter. This is a special case of the models considered in this paper. Such models are named as “semiparametric models”, since a distribution in those is characterized through a finite parameter (of interest), and an infinite dimensional parameter which captures all characteristics of the distribution except the finite dimensional one; here the model is also defined in terms of regularity and the aim of the statistical estimation is to recover estimates of the parameter of interest taking into account the regularity assumption, and also to provide the estimate of the assumed density of the generating distribution of the data.

How can statistical criterions handle the complexity of such a context, taking into account the specificity of the infinite dimensional part of the description of the model? Or phrasing differently, which is a reasonable description of the model (in terms of regularity, or other) which still makes inference on the finite dimensional parameter feasible through standard parametric inferential tools, and how should the practical inferential procedure be defined?

We consider a special class of divergences, whose properties are well known in the pure parametric setting (the power class of divergences introduces by Basu et al. [1998] and consider semi parametric models which can be worked out through this class of criterions, taking into account of the analytic properties of the model.