Asymptotic distribution of a Cramér-von mises type statistic for testing symmetry when the center is estimated

  • Sigeo Aki


In this paper we investigate the effect of estimating the center of symmetry on a Cramér-von Mises type statistic for testing the symmetry of a distribution function. The test statistic is defined by\(nT_0 \left[ {F_n } \right] = n\int_\infty ^\infty {\left\{ {F_n \left( x \right) + Fn\left( {2S\left[ {F_n } \right] - x} \right) - 1} \right\}^2 } dF_n \left( x \right),\) whereFn is the empirical distribution function andS[Fn] is an estimator of the center ofF which is consistent with the ordern1/2 and has von Mises derivative. The asymptotic distribution ofnT0[Fn] under the null hypothesis is obtained. The distribution depends on the distributionF and on the estimatorS[Fn].


Null Hypothesis Asymptotic Distribution Empirical Distribution Function Point Versus Brownian Bridge 


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© The Institute of Statistical Mathematics, Tokyo 1981

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  • Sigeo Aki

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