On modified Anderson-Darling test statistics with asymptotic properties

Abstract

This study proposes two modified Anderson-Darling test statistics that emphasize the upper or lower tails of distribution. Under the finite sample sizes, various models are used to estimate the upper tail percentage points of the proposed test statistic. The limiting distribution of the proposed test statistics is derived with a sum of infinite series. The polynomial approximation, the method of Durbin and Knott and the normal and non normal saddlepoint approximations are used to approximate the limiting distribution of the suggested test statistics. Simulations are used to investigate the accuracy of various approximations. The asymptotic power of the proposed statistics is obtained to compare with various statistics. The proposed test statistics are illustrated by the analysis of real data.