Estimation de quantiles géométriques conditionnels et non conditionnels

Chaouch, Mohamed; Gannoun, Ali; Saracco, Jérôme

Chaouch, Mohamed ; Gannoun, Ali ; Saracco, Jérôme

Journal de la société française de statistique, Tome 150 (2009) no. 2, pp. 1-27.

Résumé
Abstract

L’absence d’un critère pour ordonner les observations représente un obstacle pour étendre la définition classique des quantiles univariés au cas multidimensionnel. Dans le cadre d’études biomédicales ou industrielles, par exemple, on cherche souvent à déterminer le quantile d’un vecteur aléatoire conditionnellement à un autre. Plusieurs définitions des quantiles (conditionnels) multivariés, ne reposant pas sur une relation d’ordre, ont été proposées dans la littérature statistique. Dans cet article, nous nous focalisons sur la notion de quantile géométrique et de quantile géométrique conditionnel, fondée sur la minimisation d’une fonction de perte.

Lack of objective basis for ordering multivariate observations is a major problem in extending the notion of quantiles in a multidimensional setting. Conditional quantiles are required in various biomedical or industrial problems. Numerous alternative definitions of (conditional) quantile for multidimensional variables, have been proposed in statistical literature. In this article, we focus on the notion of geometric quantile and conditional geometric quantile, based on the minimization of a loss function.

Zbl

Classification : 62G05, 62H11, 62G20
Mot clés : algorithmes de calcul, estimateur à noyau, contours, quantile géométrique, quantile géométrique conditionnel, Transformation-Retransformation
Keywords: algorithm, geometric quantile, conditional geometric quantile, kernel estimator, contour plot, tranformation-retransformation estimate

@article{JSFS_2009__150_2_1_0,
     author = {Chaouch, Mohamed and Gannoun, Ali and Saracco, J\'er\^ome},
     title = {Estimation de quantiles g\'eom\'etriques conditionnels et non conditionnels},
     journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique},
     pages = {1--27},
     publisher = {Soci\'et\'e fran\c{c}aise de statistique},
     volume = {150},
     number = {2},
     year = {2009},
     zbl = {1311.62042},
     language = {fr},
     url = {http://archive.numdam.org/item/JSFS_2009__150_2_1_0/}
}

TY  - JOUR
AU  - Chaouch, Mohamed
AU  - Gannoun, Ali
AU  - Saracco, Jérôme
TI  - Estimation de quantiles géométriques conditionnels et non conditionnels
JO  - Journal de la société française de statistique
PY  - 2009
SP  - 1
EP  - 27
VL  - 150
IS  - 2
PB  - Société française de statistique
UR  - http://archive.numdam.org/item/JSFS_2009__150_2_1_0/
LA  - fr
ID  - JSFS_2009__150_2_1_0
ER  -

%0 Journal Article
%A Chaouch, Mohamed
%A Gannoun, Ali
%A Saracco, Jérôme
%T Estimation de quantiles géométriques conditionnels et non conditionnels
%J Journal de la société française de statistique
%D 2009
%P 1-27
%V 150
%N 2
%I Société française de statistique
%U http://archive.numdam.org/item/JSFS_2009__150_2_1_0/
%G fr
%F JSFS_2009__150_2_1_0

Chaouch, Mohamed; Gannoun, Ali; Saracco, Jérôme. Estimation de quantiles géométriques conditionnels et non conditionnels. Journal de la société française de statistique, Tome 150 (2009) no. 2, pp. 1-27. http://archive.numdam.org/item/JSFS_2009__150_2_1_0/

Bibliographie
Cité par

[1] Abdous, B.; Theodorescu, R. Note on the geometric quantile of a random vector, Statistics and Probability Letters, Volume 13 (1992), pp. 333-336 | Zbl

[2] Bahadur, R. R. A note on quantiles in large samples, The Annals of Mathematical Statistics, Volume 37 (1966), pp. 577-580 | Zbl

[3] Barnett, V. The ordering of multivariate data, Journal of the Royal Statistical Society, Ser. A, Volume 139 (1976), pp. 318-354

[4] Brown, B. M.; Hettmansperger, T. P. Affine invariant rank methods in the bivariate location model, Journal of the Royal Statistical Society, Ser. B, Volume 49 (1987), pp. 301-310 | Zbl

[5] Brown, B. M.; Hettmansperger, T. P. An affine invariant bivariate version of the sign test, Journal of the Royal Statistical Society, Ser. B, Volume 51 (1989), pp. 117-125 | Zbl

[6] Babu, G. J.; Rao, C. R. Joint asymptotic distribution of marginal quantile functions in samples from a multivariate population, Journal of Multivariate Analysis, Volume 27 (1988), pp. 15-23 | Zbl

[7] Brown, B. M. Statistical use of the spatial median, Journal of the Royal Statistical Society, Ser. B, Volume 45 (1983), pp. 25-30 | Zbl

[8] Bedall, F.K.; Zimmermann, H. Algorithm AS 143, the Mediancenter, Applied Statistics, Volume 28 (1979), pp. 325-328 | Zbl

[9] Chakraborty, B.; Chaudhuri, P. On a transformation and retransformation technique for constructing an affine equivariant multivariate median, Proceeding of the American Mathematical Society, Volume 124 (1996), pp. 2539-2547 | Zbl

[10] Cheng, Y.; G., De Gooijer J. On the uth geometric conditional quantile, Journal of Statistical Planning and Inference, Volume 137 (2007), pp. 1914-1930 | Zbl

[11] Chakraborty, B. On affine equivariant multivariate quantiles, The Institute of Statistical Mathematics, Volume 53 (2001), pp. 380-403 | Zbl

[12] Chaudhuri, P. Multivariate location estimation using extension of $R$ -estimates through $U$ -statistics type approach, The Annals of Statistics, Volume 20 (1992), pp. 897-916 | Zbl

[13] Chaudhuri, P. On a geometric notation of quantiles for multivariate data, Journal of the American Statistical Association, Volume 91 (1996), pp. 862-872 | Zbl

[14] Chaudhuri, P.; Sengupta, D. Sign tests in multidimension : inference based on the geometry of the data cloud, Journal of the American Statistical Association, Volume 88 (1993), pp. 1363-1370 | Zbl

[15] Donoho, D. L.; Gasko, M. Breakdown properties of location estimates based on halfspace depth and projected outlyingness, The Annals of Statistics, Volume 20 (1992), pp. 1803-1827 | Zbl

[16] De Gooijer, J. G.; Gannoun, A. TR multivariate conditional median estimation, Communications in Statistics - Simulation and Computation, Volume 36 (2007), pp. 165-176 | Zbl

[17] De Gooijer, J. G.; Gannoun, A.; Zerom, D. Mean squared error properties of kernel-based multi-stage conditional median predictor for time series, Statistics and Probablility Letters, Volume 56 (2002), pp. 51-56 | Zbl

[18] De Gooijer, J. G.; Gannoun, A.; Zerom, D. A multivariate quantile predictor, Communications in Statistics - Theory and Methods, Volume 35 (2006), pp. 133-147 | Zbl

[19] Eddy, W.F. Convex Hull Peeling, COMPSTAT 1982 for IASC, Vienna : Pysica-Verlag (1982), pp. 42-47 | Zbl

[20] Eddy, W.F. Ordering of Multivariate Data., Computer Science and Statistics : The Interface. L. Billard, Amesterdam : North-Holland (1985), pp. 25-30

[21] Ferguson, T. Mathematical Statistics : A Decision Theoric Approach, Academic Press, New York, 1967 | Zbl

[22] Gannoun, A.; Girard, S.; Guinot, C.; Saracco, J. Trois méthodes non paramétriques pour l’estimation de courbes de référence. Application à l’analyse des propriétés biophysiques de la peau, Revue de Statistique Appliquée, Volume 1 (2002), pp. 65-89

[23] Gower, J.C. Algorithm AS 78 : The Mediancenter, Applied Statistics, Volume 23 (1974), pp. 466-470

[24] Gannoun, A.; Saracco, J.; Yu, K. Nonparametric time series prediction by conditional median and quantiles, Journal of Statistical Planning and Inference, Volume 117 (2003), pp. 207-223 | Zbl

[25] Gannoun, A.; Saracco, J.; Yan, A.; Bonney, G.E. On adaptive transformation-retransformation estimate of conditional spatial median, Communications in Statistics - Theory and Methods, Volume 32 (2003), pp. 1981-2011 | Zbl

[26] Haldane, J. B. S. Note on the median of a multivariate distribution, Biometrika, Volume 35 (1948), pp. 414-415 | Zbl

[27] Koenker, R.; Basset, G. Regression quantiles, Econometrica, Volume 46 (1978), pp. 33-50 | Zbl

[28] Kokic, P.; Breckling, J.; Lübke, O. A new definition of multivariate $M$ -quantiles, Statistical data analysis based on the $L_{1}$ -norm and related methods, Birkhäuser Verlag, Basel (2002), pp. 15-24 | Zbl

[29] Kemperman, J. H. B. The median of a finite measure on a Banach space, Statistical Data Analysis based on the $L_{1}$ -norm and related methods, Y. Dodge (ed.), North-Holland, Amsterdam (1987), pp. 217-230

[30] Koltchinskii, V. M-estimation, convexity and quantiles, The Annals of Statistics, Volume 25 (1997), pp. 435-477 | Zbl

[31] Liu, R. Y.; Parelius, J. M.; Singh, K. Multivariate analysis by data depth : descriptive statistics, graphics and inference (with discussion), The Annals of Statistics, Volume 27 (1999), pp. 783-858 | Zbl

[32] Oja, H. Descriptive statistics for multivariate trimming, Statistics and Probability Letters, Volume 1 (1983), pp. 327-332 | Zbl

[33] Plackett, R. L. Comment on the “ Ordering of multivariate data”, by V. Barnett, Journal of the Royal Statistical Society, Ser. A, Volume 139 (1976), pp. 344-346

[34] Reiss, R. D. Approximate distributions of order statistics with applications to nonparametric statistics, New York : Springer, 1989 | MR | Zbl

[35] Serfling, R. Quantile functions for multivariate analysis : approaches and applications, Statistica Neerlandica, Volume 56 (2002), pp. 214-232 | MR | Zbl

[36] Serfling, R. Nonparametric multivariate descriptive measures based on spatial quantiles, Journal of Statistical Planning and Inference, Volume 123 (2004), pp. 259-278 | MR | Zbl

[37] Zuo, Y.; Serfling, R. General notions of statistical depth function, The Annals of Statistics, Volume 28 (2000), pp. 461-482 | MR | Zbl