Optimal nonlinear transformations of random variables
Goia, Aldo ; Salinelli, Ernesto
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010), p. 653-676 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

Publié le : 2010-01-01
DOI : https://doi.org/10.1214/09-AIHP326
Classification:  60E05,  49J05,  47A75,  62G05,  62G10 
@article{AIHPB_2010__46_3_653_0,
     author = {Goia, Aldo and Salinelli, Ernesto},
     title = {Optimal nonlinear transformations of random variables},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {46},
     year = {2010},
     pages = {653-676},
     doi = {10.1214/09-AIHP326},
     mrnumber = {2682262},
     zbl = {1201.62077},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2010__46_3_653_0}
}

Goia, Aldo; Salinelli, Ernesto. Optimal nonlinear transformations of random variables. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) pp. 653-676. doi : 10.1214/09-AIHP326. http://gdmltest.u-ga.fr/item/AIHPB_2010__46_3_653_0/

[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. | MR 2091081 | Zbl pre05058921

[2] G. Arfken. Mathematical Methods for Physicists. Academic Press, New York, 1966. | MR 205512 | Zbl pre05948059

[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. | MR 1154349 | Zbl 0737.62032

[4] P. Burman. Rates of convergence for the estimate of the optimal transformations of variables. Ann. Statist. 19 (1991) 702-723. | MR 1105840 | Zbl 0733.62054

[5] P. Burman and K. W. Chen. Nonparametric estimation of a regression function. Ann. Statist. 17 (1989) 1567-1596. | MR 1026300 | Zbl 0744.62054

[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. | MR 1694383 | Zbl 0915.49001

[7] T. Cacoullos. On upper and lower-bounds for the variance of a function of a random variable. Ann. Probab. 10 (1982) 799-809. | MR 659549 | Zbl 0492.60021

[8] T. Cacoullos and V. Papathanasiou. On upper bounds for the variance of functions of random variables. Statist. Probab. Lett. 3 (1985) 175-184. | MR 801687 | Zbl 0572.60021

[9] T. Cacoullos and V. Papathanasiou. Characterizations of distributions by variance bounds. Statist. Probab. Lett. 7 (1989) 351-356. | MR 1001133 | Zbl 0677.62012

[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. | MR 1491576 | Zbl 0922.62009

[11] H. Cardot. Spatially adaptive splines for statistical linear inverse problems. J. Multivariate Anal. 81 (2002) 100-119. | MR 1901208 | Zbl 1005.65053

[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. | Numdam | MR 877386 | Zbl 0612.60013

[13] H. Chernoff. A note on an inequality involving the normal distribution. Ann. Probab. 9 (1981) 533-535. | MR 614640 | Zbl 0457.60014

[14] R. Courant and D. Hilbert. Methods of Mathematical Physics. Wiley, New York, 1989. | MR 1013360 | Zbl 0729.00007

[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. | MR 650934 | Zbl 0539.62064

[16] C. De Boor. A Practical Guide to Splines. Springer, New York, 2001. | MR 1900298 | Zbl 0987.65015

[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. | MR 1668907 | Zbl 0927.62061

[18] I. M. Gelfand and S. V. Fomin. Calculus of Variations. Prentice-Hall, New Jersey, 1963. | MR 160139 | Zbl 0127.05402

[19] P. Gurka and B. Opic. Continuous and compact imbeddings of weighted Sobolev Spaces II. Czechoslovak Math. J. 39 (1989) 78-94. | MR 983485 | Zbl 0669.46019

[20] C. A. J. Klaassen. On an inequality of Chernoff. Ann. Probab. 13 (1985) 966-974. | MR 799431 | Zbl 0576.60015

[21] A. Kufner and B. Opic. How to define reasonably weighted Sobolev Spaces. Comment. Math. Univ. Carolin. 25 (1984) 537-554. | MR 775568 | Zbl 0557.46025

[22] O. Johnson and A. Barron. Fisher information inequalities and the central limit theorem, Probab. Theory Related Fields 129 (2004) 391-409. | MR 2128239 | Zbl 1047.62005

[23] I. T. Jolliffe. Principal Component Analysis. Springer, Berlin, 2004. | MR 841268 | Zbl 1011.62064

[24] H. O. Lancaster. The Chi-Squared Distribution. Wiley, New York, 1969. | MR 253452 | Zbl 0193.17802

[25] M.-L. T. Lee. Properties and applications of the Samarov family of bivariate distributions. Comm. Statist. Theory Methods 25 (1996) 1207-1222. | MR 1394279 | Zbl 0875.62205

[26] D. D. Mari and S. Kotz. Correlation and Dependence. Imperial College Press, London, 2001. | MR 1835042 | Zbl 0977.62004

[27] M. Okamoto. Distinctness of the eigenvalues of a quadratic form in a multivariate sample. Ann. Statist. 1 (1973) 763-765. | MR 331643 | Zbl 0261.62043

[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. | MR 395268 | Zbl 0301.65063

[29] S. Purkayastha and S. K. Bhandari. Characterization of uniform distributions by inequality of Chernoff-type, Sankhyā 52 (1990) 376-382. | MR 1178045 | Zbl 0727.62021

[30] E. Salinelli. Nonlinear principal components I. Absolutely continuous random variables with positive bounded densities. Ann. Statist. 26 (1998) 596-616. | MR 1626079 | Zbl 0929.62067

[31] E. Salinelli. Nonlinear principal components II. Characterization of normal distributions. J. Multivariate Anal. 100 (2009) 652-660. | MR 2478188 | Zbl 1169.62058

[32] L. L. Schumaker. Spline Functions: Basic Theory. Wiley, New York, 1981. | MR 606200 | Zbl 0449.41004

[33] C. J. Stone. Optimal global rate of convergence for nonparametric regression. Ann. Statist. 10 (1982) 1040-1053. | MR 673642 | Zbl 0511.62048

[34] A. Zettl. Sturm-Liouville Theory. Mathematical Survey and Monographs 121. Amer. Math. Soc., Providence, RI, 2005. | MR 2170950 | Zbl 1103.34001