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

Summary

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$nT_0 \left[ {F_n } \right] = n\int_\infty ^\infty {\left\{ {F_n \left( x \right) + Fn\left( {2S\left[ {F_n } \right] - x} \right) - 1} \right\}^2 } dF_n \left( x \right),$ whereF _n is the empirical distribution function andS[F _n] is an estimator of the center ofF which is consistent with the ordern ^1/2 and has von Mises derivative. The asymptotic distribution ofnT ₀[F_n] under the null hypothesis is obtained. The distribution depends on the distributionF and on the estimatorS[F _n].