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Biometrika 1987 74(1):177-184; doi:10.1093/biomet/74.1.177
© 1987 by Biometrika Trust
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Testing for symmetry

SÅNDOR CSÖRGO and C. R. HEATHCOTE

Bolyai Institute, Szeged Univeristy 6720 Szeged, Hungary
Department of Statistics, Australian National University Canberra, ACT 2601, Australia

We introduce the characteristic symmetry function, based on the characteristic function of the underlying distribution, whose behaviour is indicative of symmetry or its absence. A statistic is proposed for testing symmetry about an unspecified centre, derived from the empirical characteristic symmetry function. The statistic is readily computible, it utilizes information in the empirical characteristic function over an interval, and does not require the estimation of the centre of symmetry. Under general symmetry the asymptotic null distribution of the statistic is folded normal. The empirical power for selected alternatives is studied by a small-scale simulation and a numerical illustration is given.

Key Words: Centre of symmetry • Characteristic symmetry function • Empirical characteristic function • Maximum and minimum limiting variance • Symmetry test • Weak convergence


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