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ON NONPARAMETRIC TESTS FOR SYMMETRY IN *R*^{m}

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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.

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