On Some Karhunen–Loève Expansions Related to SU(2) with Application to Tests of Symmetry
- 1Laboratoire LaMME (UMR 8071), Department of Mathematics, University of Evry Paris-Saclay, 91000 Evry-Courcouronnes, France
Abstract
Karhunen–Loève expansions play a key role in mathematical statistics. Two kernels associated with the well-known Anderson–Darling statistic admit such developments involving Legendre polynomials and their derivatives. We give an interpretation of these developments in terms of special functions occurring as the matrix elements of irreducible unitary representations of the compact Lie group SU(2). This interpretation leads us to introduce several families of bilinear expansions defined over SU(2)×SU(2). In this framework, the Anderson–Darling test for uniformity has an interpretation in terms of rotational invariance of some probability distributions. We provide a similar geometric interpretation of our new developments. We outline statistical applications.
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BibTeX
@article{pycke2026some,
title = {{On Some Karhunen–Loève Expansions Related to SU(2) with Application to Tests of Symmetry}},
author = {Jean-Renaud Pycke},
journal = {Mathematics},
year = {2026},
month = nov,
doi = {10.3390/math14214041},
url = {https://www.mdpi.com/2227-7390/14/21/4041},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - On Some Karhunen–Loève Expansions Related to SU(2) with Application to Tests of Symmetry
AU - Jean-Renaud Pycke
PY - 2026
DA - 2026/11/04
JO - Mathematics
DO - 10.3390/math14214041
UR - https://www.mdpi.com/2227-7390/14/21/4041
AB - Karhunen–Loève expansions play a key role in mathematical statistics. Two kernels associated with the well-known Anderson–Darling statistic admit such developments involving Legendre polynomials and their derivatives. We give an interpretation of these developments in terms of special functions occurring as the matrix elements of irreducible unitary representations of the compact Lie group SU(2). This interpretation leads us to introduce several families of bilinear expansions defined over SU(2)×SU(2). In this framework, the Anderson–Darling test for uniformity has an interpretation in terms of rotational invariance of some probability distributions. We provide a similar geometric interpretation of our new developments. We outline statistical applications.
SN - 2227-7390
ER -