Optimal nonlinear transformations of random variables
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 3, pp. 653-676.

Dans cet article nous étudions les composantes principales non linéaires définies par Salinelli en 1998, dans le cas d'une variable aléatoire réelle. La signification probabiliste et statistique est approfondie et des propriétés sont illustrées. Une procédure d'estimation basée sur les fonctions splines, qui adapte la méthode classique de Rayleigh-Ritz, est présentée. Des propriétés asymptotiques de cet estimateur sont établies, et on donne une borne pour la vitesse de convergence sous des conditions générales. Des applications aux tests d'ajustement et à la construction de distributions bivariées sont proposées.

In this paper we deepen the study of the nonlinear principal components introduced by Salinelli in 1998, referring to a real random variable. New insights on their probabilistic and statistical meaning are given with some properties. An estimation procedure based on spline functions, adapting to a statistical framework the classical Rayleigh-Ritz method, is introduced. Asymptotic properties of the estimator are proved, providing an upper bound for the rate of convergence under suitable mild conditions. Some applications to the goodness-of-fit test and the construction of bivariate distributions are proposed.

DOI : 10.1214/09-AIHP326
Classification : 60E05, 49J05, 47A75, 62G05, 62G10
Mots-clés : covariance operator, Chernoff-Poincaré inequality, nonlinear principal components, splines estimates, Sturm-Liouville problems 
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Goia, Aldo; Salinelli, Ernesto. Optimal nonlinear transformations of random variables. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 3, pp. 653-676. doi : 10.1214/09-AIHP326. http://www.numdam.org/articles/10.1214/09-AIHP326/

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