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Robust inference based on flexible parametric families of distributions

We discuss an approach to robustness based on the adoption of a flexible parametric family of distributions as a probability model of the observations. A sensible general requirement for this class of distributions is that it includes parameters to regulate asymmetry and kurtosis, in addition to location and scale; in specific situations, it may be appropriate to consider classes of distributions with more detailed features. Within this framework, standard inferential procedures can be used, whether in the classical or in the Bayesian paradigm. The need to develop specific robust techniques is removed by the ability of the adopted family of distributions to peek up a member which adapts itself to the empirical features of the data. The key conceptual difference of such formulation with respect to the traditional robust approach is in the explicit specification of a probability model, and so in the clearer identification of the quantities which are the target of the inference process. Hence, the quantities to be estimated are explicitly defined as the parameters of the parent distribution, while an M-estimator, say, converges asymptotically to the solution of a non-linear equation, that is a quantity which is in general explicitly known only if the error distribution is symmetric. Indeed the vast majority of robustness studies has focussed on the case of symmetric distributions. There is however convincing empirical evidence that data in real applications do not conform to the symmetry assumption; see for instance Hill & Dixon (1982). In addition, the existence of a probability model allows simple estimation of quantities not inherent to the inference process, for instance the tail probability Pr{X > t} when this is not a parameter of the fitting process, while M-estimators do not provide a similar feature. The implementation of the above-described paradigm requires the specification of a suitably flexible class of distributions. One such proposal is the skew-exponential power distribution proposed in Azzalini (1986) and studied in detail by DiCiccio & Monti (2004). A different but connected specification is based on the skew-t distribution, which has been examined in this role by Azzalini & Genton (2008). The talk will illustrate this formulation, and it will discuss various aspects and problems connected.