New Karhunen-Loève expansions based on Hahn polynomials with application to a Sobolev test for uniformity on Johnson graphs

Abstract

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.

Keywords

• Karhunen-Loève expansions
• classical orthogonal polynomials
• distance regular graphs
• directional statistics