On Some Karhunen–Loève Expansions Related to SU(2) with Application to Tests of Symmetry

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 $SU\left(2\right)$. This interpretation leads us to introduce several families of bilinear expansions defined over $SU\left(2\right)\times SU\left(2\right)$. 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.