Optimal nonlinear transformations of random variables

Aldo Goia; Ernesto Salinelli

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 3, page 653-676
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Goia, Aldo, and Salinelli, Ernesto. "Optimal nonlinear transformations of random variables." Annales de l'I.H.P. Probabilités et statistiques 46.3 (2010): 653-676. <http://eudml.org/doc/240675>.

@article{Goia2010,
abstract = {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.},
author = {Goia, Aldo, Salinelli, Ernesto},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {covariance operator; Chernoff–Poincaré inequality; nonlinear principal components; splines estimates; Sturm–Liouville problems; Chernoff-Poincaré inequality; spline estimates; Sturm-Liouville problems},
language = {eng},
number = {3},
pages = {653-676},
publisher = {Gauthier-Villars},
title = {Optimal nonlinear transformations of random variables},
url = {http://eudml.org/doc/240675},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Goia, Aldo
AU - Salinelli, Ernesto
TI - Optimal nonlinear transformations of random variables
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 3
SP - 653
EP - 676
AB - 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.
LA - eng
KW - covariance operator; Chernoff–Poincaré inequality; nonlinear principal components; splines estimates; Sturm–Liouville problems; Chernoff-Poincaré inequality; spline estimates; Sturm-Liouville problems
UR - http://eudml.org/doc/240675
ER -

References

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