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By Geert Dhaene

The heritage of many sciences is characterised by means of a virtually non-stop emer­ gence of latest theories. From a normative standpoint, the survival of a brand new conception should still commonly be made up our minds through its skill to provide an explanation for a brand new physique of proof which the present theories are not able to provide an explanation for. If furthermore the hot concept is ready to clarify all of the effects bought by way of the present theories and if it could actually indicate why those theories fail to give an explanation for definite proof, it may develop into the dominant idea. another way, it could coexist with different theories for a while. for this reason, a brand new conception needs to be judged not just through confronting it with current proof, but additionally by way of confronting it with present theories. the concept that a conception will be in a position to account for the consequences ob­ tained by way of different theories, even if implicitly adhered to through many scientists, has not often been formalized. The information literature on parametric hypoth­ esis trying out, notwithstanding, will be visible for instance of one of these formalization.

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Example text

O Since extensions are non-unique and highly arbitrary, it may be advisable to apply only reductions to modify the distributions of the models. Furthermore, considering the Kullback-Leibler distance between the true distribution of a set of variables and a model thereof, it is easily seen that an extension of the distributions of the model adds a non-negative term to the distance. This term is zero if and only if the extended part is correctly specified. Otherwise said, an extension can only render the model 'more misspecified'.

A] ILl ) ' provided that 2a has full column rank, which requires T 2: n. 28) by reversing the roles of:F and g. The conditional mean and variance of the pseudo-true distribution now have interpretations in terms of orthogonal projections in the Euclidean space RT, equipped with the usual norm and inner product. Let F~ (resp. Gil) be the product probability measure induced by the measures Fo: (resp. G(3), given X = Xt, t = 1, ... , T. Then, F~ (resp. LI (resp. SGf3d and conditional covariance matrix a 2I (resp.

2. Looking at the different approaches described above, it is a natural question whether the robust approach can be extended to non-nested hypothesis testing. 10) and without assuming that :F c g. As a matter of fact, this formulation disconnects a hypothesis on the pseudo-true values from the maintained hypothesis and drops the latter. As we will show later on, it is possible to construct tests which are consistent against any departure from He, provided that certain regularity conditions hold.

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