On a Karhunen-Lo\`eve expansion based on Krawtchouk polynomials with application to Bahadur optimality for the binomial location family

Abstract

In this paper we extend to the discrete case a Karhunen-Lo\`eve expansion already known for continuous families of classical orthogonal polynomials. This expansion involves Krawtchouk polynomials. It provides us with the orthogonal decomposition of the covariance function of a weighted discrete Brownian bridge process. We introduce a discrete Cram\'er-vonMises statistic associated with this covariance function. We show that this statistic satisfies a property of Bahadur local optimality for a statistical test in the location family for binomial distributions. Our statistic and the goodness-of-fit problem we deal with can be seen as a discrete version of a problem stated by Y. Nikitin about the statistic of de Wet and Venter. Our proofs make use of the formulas valid for all classical orthogonal families of polynomials, so that the way most of our results can be extended to Meixner, Hahn, and Charlier polynomials and the associated distributions is clearly outlined.