On the use of reproducing kernel Hilbert spaces in functional classification

The H\'ajek-Feldman dichotomy establishes that two Gaussian measures are either mutually absolutely continuous with respect to each other (and hence there is a Radon-Nikodym density for each measure with respect to the other one) or mutually singular. Unlike the case of finite dimensional Gaussian measures, there are non-trivial examples of both situations when dealing with Gaussian stochastic processes. This paper provides: (a) Explicit expressions for the optimal (Bayes) rule and the minimal classification error probability in several relevant problems of supervised binary classification of mutually absolutely continuous Gaussian processes. The approach relies on some classical results in the theory of Reproducing Kernel Hilbert Spaces (RKHS). (b) An interpretation, in terms of mutual singularity, for the "near perfect classification" phenomenon described by Delaigle and Hall (2012). We show that the asymptotically optimal rule proposed by these authors can be identified with the sequence of optimal rules for an approximating sequence of classification problems in the absolutely continuous case. (c) A new model-based method for variable selection in binary classification problems, which arises in a very natural way from the explicit knowledge of the RN-derivatives and the underlying RKHS structure. Different classifiers might be used from the selected variables. In particular, the classical, linear finite-dimensional Fisher rule turns out to be consistent under some standard conditions on the underlying functional model

On the maximum bias functions of MM-estimates and constrained M-estimates of regression

We derive the maximum bias functions of the MM-estimates and the constrained M-estimates or CM-estimates of regression and compare them to the maximum bias functions of the S-estimates and the $\tau$-estimates of regression. In these comparisons, the CM-estimates tend to exhibit the most favorable bias-robustness properties. Also, under the Gaussian model, it is shown how one can construct a CM-estimate which has a smaller maximum bias function than a given S-estimate, that is, the resulting CM-estimate dominates the S-estimate in terms of maxbias and, at the same time, is considerably more efficient.Comment: Published at http://dx.doi.org/10.1214/009053606000000975 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Between Honour and Excellence: Nobiliary Genealogy and Common Opinion in Early Modern Spain

This article inquires after the causes of the unprecedented growth and scope of genealogical expertise in the many realms that comprised the Spanish Monarchy in the seventeenth century. Lengthy proofs of nobility were a prerequisite for admission to orders of chivalry, courtly institutions, colleges, and universities. The nature and means of transmission of genealogical knowledge are analysed in order to grasp its socio-political significance. Indeed, besides their critical importance for the nobility, genealogies were relevant for society at large and were tied to the recurring debates on the essence of nobility that were taking place in Europe from the thirteenth century.This article inquires after the causes of the unprecedented growth and scope of genealogical expertise in the many realms that comprised the Spanish Monarchy in the seventeenth century. Lengthy proofs of nobility were a prerequisite for admission to orders of chivalry, courtly institutions, colleges, and universities. The nature and means of transmission of genealogical knowledge are analysed in order to grasp its socio-political significance. Indeed, besides their critical importance for the nobility, genealogies were relevant for society at large and were tied to the recurring debates on the essence of nobility that were taking place in Europe from the thirteenth century