We give a new family of Karhunen-Loève expansions involving Hahn polynomials. This enables us to introduce discrete analogues of Watson statistics, and a test for uniformity on Johnson’s graphs. We use the fact that the zonal spherical functions on these graphs are Hahn polynomials. Our test is consistent against all alternatives and locally most powerful against some alternative.
Nous proposons un nouveau développement de Karhunen-Loève avec les polynômes de Hahn. Cela nous permet d’introduire une analogue discrète de la statistique de Watson pour tester l’uniformité d’une distribution sur les graphes de Johnson, dont les fontions sphériques zonales sont les polynômes de Hahn. Ce test peut être vu comme un test de Sobolev dans le cas discret, nous en déduisons certaines de ses propriétés asymptotiques.
Accepted:
Published online:
DOI: 10.5802/crmath.470
Keywords: Karhunen-Loève expansions, classical orthogonal polynomials, distance regular graphs, directional statistics
Mots-clés : développements de Karhunen-Loève, polynômes orthogonaux classiques, Graphes distance-réguliers, statistiques directionnelles
Jean-Renaud Pycke 1
CC-BY 4.0
@article{CRMATH_2024__362_G7_701_0,
author = {Jean-Renaud Pycke},
title = {New {Karhunen-Lo\`eve} expansions based on {Hahn} polynomials with application to a {Sobolev} test for uniformity on {Johnson} graphs},
journal = {Comptes Rendus. Math\'ematique},
pages = {701--711},
publisher = {Acad\'emie des sciences, Paris},
volume = {362},
year = {2024},
doi = {10.5802/crmath.470},
mrnumber = {4805498},
language = {en},
}
TY - JOUR AU - Jean-Renaud Pycke TI - New Karhunen-Loève expansions based on Hahn polynomials with application to a Sobolev test for uniformity on Johnson graphs JO - Comptes Rendus. Mathématique PY - 2024 SP - 701 EP - 711 VL - 362 PB - Académie des sciences, Paris DO - 10.5802/crmath.470 LA - en ID - CRMATH_2024__362_G7_701_0 ER -
%0 Journal Article %A Jean-Renaud Pycke %T New Karhunen-Loève expansions based on Hahn polynomials with application to a Sobolev test for uniformity on Johnson graphs %J Comptes Rendus. Mathématique %D 2024 %P 701-711 %V 362 %I Académie des sciences, Paris %R 10.5802/crmath.470 %G en %F CRMATH_2024__362_G7_701_0
Jean-Renaud Pycke. New Karhunen-Loève expansions based on Hahn polynomials with application to a Sobolev test for uniformity on Johnson graphs. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 701-711. doi : 10.5802/crmath.470. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.470/
[1] Testing for uniformity on a compact homogeneous space, J. Appl. Probab., Volume 5 (1968), pp. 177-195 | DOI | MR | Zbl
[2] Algebraic graph theory, Cambridge Mathematical Library, Cambridge University Press, 1994 | MR
[3] Modern graph theory, Graduate Texts in Mathematics, 184, Springer, 1998 | DOI | MR
[4] Distance-Regular Graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 18, Springer, 1989 | DOI | MR
[5] Spectra of Graphs: Theory and Applications, J. A. Barth Verlag, 1998 | MR
[6] Goodness-of-Fit Techniques (Ralph B. D’Agostino; Michael A. Stephens, eds.), Statistics: Textbooks and Monographs, 68, Marcel Dekker, 1986 | MR
[7] Distribution theory for tests based on the sample distribution function, CBMS-NSF Regional Conference Series in Applied Mathematics, 9, Society for Industrial and Applied Mathematics, 1973 | DOI | MR
[8] Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev norms, Ann. Stat., Volume 3 (1975) no. 6, pp. 1243-1266 | MR | Zbl
[9] Continuous univariate distributions. Volume 2, John Wiley & Sons, 1995
[10] Representations of Lie groups and special functions. Volume II, Kluwer Academic Publishers, 1995
[11] Hypergeometric orthogonal polynomials and their -analogues, Springer Monographs in Mathematics, Springer, 2010 | DOI | MR
[12] Theory of -statistics, Springer, 2013
[13] Directional statistics, Wiley Series in Probability and Statistics, John Wiley & Sons, 2000 | MR
[14] Classical orthogonal polynomials of a discrete variable, Springer Series in Computational Physics, Springer, 1991 | DOI
[15] Recent advances in directional statistics, Test, Volume 30 (2021) no. 1, pp. 1-58 | DOI | MR | Zbl
[16] U-statistics based on the Green’s function of the Laplacian on the circle and the sphere, Stat. Probab. Lett., Volume 77 (2007) no. 9, pp. 863-872 | DOI | MR
[17] Approximation theorems of mathematical statistics, Wiley Series in Probability and Statistics, John Wiley & Sons, 2009 | MR
[18] Empirical processes with applications to statistics, Classics in Applied Mathematics, 59, Society for Industrial and Applied Mathematics, 2009 | DOI | MR
[19] An introduction to group representations and orthogonal polynomials, Orthogonal polynomials: theory and practice (NATO ASI Series. Series C. Mathematical and Physical Sciences), Volume 294, Kluwer Academic Publishers, 1990, pp. 419-433 | DOI | Zbl
[20] Asymptotic statistics, Cambridge Series in Statistical and Probabilistic Mathematics, 3, Cambridge University Press, 2000 | MR
[21] Goodness-of-fit tests on a circle, Biometrika, Volume 48 (1961) no. 1/2, pp. 109-114 | DOI | MR | Zbl
Cited by Sources:
Comments - Policy