This paper gives an estimate of a symmetric population distribution function Fand a modified K. Pearson statistic in the symmetric auxiliary information. Asymptotical properties show that they dominate the usual statistics. The asymptotic distribution of the modified K. Pearson statistic under the null hypothesis is derived. The approximate Bahadur slope and asymptotic power function of a test based on the modified K. Pearson statistic are given. Numerical results show the modified K. Pearson statistic is better than K. Pearson statistic.