Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Optimal nonlinear transformations of random variables

Aldo Goia and Ernesto Salinelli

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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.


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

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

First available in Project Euclid: 6 August 2010

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Zentralblatt MATH identifier

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

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


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.

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  • [1] F. Antoci. Some necessary and some sufficient conditions for the compactness of the embedding of weighted Sobolev spaces. Ric. Mat. LII (2003) 55–71.
  • [2] G. Arfken. Mathematical Methods for Physicists. Academic Press, New York, 1966.
  • [3] D. Bosq. Modelization, nonparametric estimation and prediction for continuous time process. In Nonparametric Functional Estimation and Related Topics 509–529. G. Roussas, (Ed.). Nato, Asi Series. Kluwer Academic, Dordrecht, 1991.
  • [4] P. Burman. Rates of convergence for the estimate of the optimal transformations of variables. Ann. Statist. 19 (1991) 702–723.
  • [5] P. Burman and K. W. Chen. Nonparametric estimation of a regression function. Ann. Statist. 17 (1989) 1567–1596.
  • [6] G. Buttazzo, M. Giaquinta and S. Hildebrandt. One-Dimensional Variational Problems. Oxford Lecture Series in Mathematics and Its Applications 15. Clarendon Press, New York, 1998.
  • [7] T. Cacoullos. On upper and lower-bounds for the variance of a function of a random variable. Ann. Probab. 10 (1982) 799–809.
  • [8] T. Cacoullos and V. Papathanasiou. On upper bounds for the variance of functions of random variables. Statist. Probab. Lett. 3 (1985) 175–184.
  • [9] T. Cacoullos and V. Papathanasiou. Characterizations of distributions by variance bounds. Statist. Probab. Lett. 7 (1989) 351–356.
  • [10] T. Cacoullos and V. Papathanasiou. Characterization of distributions by generalizations of variance bounds and simple proofs of the CLT. J. Statist. Plann. Inference 63 (1997) 157–171.
  • [11] H. Cardot. Spatially adaptive splines for statistical linear inverse problems. J. Multivariate Anal. 81 (2002) 100–119.
  • [12] L. H. Y. Chen and J. H. Lou. Characterization of probability distributions by Poincaré-type inequalities. Ann. Inst. H. Poincaré Probab. Statist 23 (1987) 91–110.
  • [13] H. Chernoff. A note on an inequality involving the normal distribution. Ann. Probab. 9 (1981) 533–535.
  • [14] R. Courant and D. Hilbert. Methods of Mathematical Physics. Wiley, New York, 1989.
  • [15] J. Dauxois, A. Pousse and Y. Romain. Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference. J. Multivariate Anal. 12 (1982) 136–154.
  • [16] C. De Boor. A Practical Guide to Splines. Springer, New York, 2001.
  • [17] N. E. El Faouzi and P. Sarda. Rates of convergence for spline estimates of additive principal components. J. Multivariate Anal. 68 (1999) 120–137.
  • [18] I. M. Gelfand and S. V. Fomin. Calculus of Variations. Prentice-Hall, New Jersey, 1963.
  • [19] P. Gurka and B. Opic. Continuous and compact imbeddings of weighted Sobolev Spaces II. Czechoslovak Math. J. 39 (1989) 78–94.
  • [20] C. A. J. Klaassen. On an inequality of Chernoff. Ann. Probab. 13 (1985) 966–974.
  • [21] A. Kufner and B. Opic. How to define reasonably weighted Sobolev Spaces. Comment. Math. Univ. Carolin. 25 (1984) 537–554.
  • [22] O. Johnson and A. Barron. Fisher information inequalities and the central limit theorem, Probab. Theory Related Fields 129 (2004) 391–409.
  • [23] I. T. Jolliffe. Principal Component Analysis. Springer, Berlin, 2004.
  • [24] H. O. Lancaster. The Chi-Squared Distribution. Wiley, New York, 1969.
  • [25] M.-L. T. Lee. Properties and applications of the Samarov family of bivariate distributions. Comm. Statist. Theory Methods 25 (1996) 1207–1222.
  • [26] D. D. Mari and S. Kotz. Correlation and Dependence. Imperial College Press, London, 2001.
  • [27] M. Okamoto. Distinctness of the eigenvalues of a quadratic form in a multivariate sample. Ann. Statist. 1 (1973) 763–765.
  • [28] J. G. Pierce and R. S. Varga. Higher order convergence results for the Rayleigh–Ritz method applied to eigenvalue problems. I: Estimates relating Rayleigh–Ritz and Galerkin approximations to eigenfunctions. SIAM J. Numer. Anal. 9 (1972) 137–151.
  • [29] S. Purkayastha and S. K. Bhandari. Characterization of uniform distributions by inequality of Chernoff-type, Sankhyā 52 (1990) 376–382.
  • [30] E. Salinelli. Nonlinear principal components I. Absolutely continuous random variables with positive bounded densities. Ann. Statist. 26 (1998) 596–616.
  • [31] E. Salinelli. Nonlinear principal components II. Characterization of normal distributions. J. Multivariate Anal. 100 (2009) 652–660.
  • [32] L. L. Schumaker. Spline Functions: Basic Theory. Wiley, New York, 1981.
  • [33] C. J. Stone. Optimal global rate of convergence for nonparametric regression. Ann. Statist. 10 (1982) 1040–1053.
  • [34] A. Zettl. Sturm–Liouville Theory. Mathematical Survey and Monographs 121. Amer. Math. Soc., Providence, RI, 2005.