Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Optimal nonlinear transformations of random variables

Aldo Goia and Ernesto Salinelli

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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.

Résumé

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 proprietés 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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 3 (2010), 653-676.

Dates
First available in Project Euclid: 6 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1281100394

Digital Object Identifier
doi:10.1214/09-AIHP326

Mathematical Reviews number (MathSciNet)
MR2682262

Zentralblatt MATH identifier
1201.62077

Subjects
Primary: 60E05: Distributions: general theory
Secondary: 49J05: Free problems in one independent variable 47A75: Eigenvalue problems [See also 47J10, 49R05] 62G05: Estimation 62G10: Hypothesis testing

Keywords
Covariance operator Chernoff–Poincaré inequality Nonlinear principal components Splines estimates Sturm–Liouville problems

Citation

Goia, Aldo; Salinelli, Ernesto. Optimal nonlinear transformations of random variables. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 3, 653--676. doi:10.1214/09-AIHP326. https://projecteuclid.org/euclid.aihp/1281100394


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