ON NONPARAMETRIC TESTS FOR SYMMETRY IN R^{m}

SIGEO AKI

Department of Mathematical Science, Faculty of Engineering Science, Osaka University, Toyonaka 560, Japan

(Received October 2, 1991; revised February 17, 1993)

Abstract. This paper considers the problem for testing symmetry of a distribution in R^{m} based on the empirical distribution function. Limit theorems which play important roles for investigating asymptotic behavior of such tests are obtained. The limit processes of the theorems are multiparameter Wiener process. Based on the limit theorems, nonparametric tests are proposed whose asymptotic distributions are functionals of a multiparameter standard Wiener process. The tests are compared asymptotically with each other in the sense of Bahadur.

Key words and phrases: Asymptotic distribution, test for symmetry, L_{1}-norm, L_{2}-norm, empirical process, central limit theorems, goodness-of-fit tests, multiparameter Wiener process, density estimator, approximate Bahadur efficiency.