Modified two-sample Anderson-Darling test statistic

Two-sample testing hypothesis has been used and discussed in several scientific fields. However, the underlying theoretical distribution cannot be assumed to have a specific parametric distribution such as a normal distribution. When it is difficult to assume an underlying distribution, non parametric statistical methods are preferable. Such methods are typically robust and require fewer assumptions regarding the underlying distribution. As it is difficult to determine the number of parameters in an unknown distribution, we are interested in testing whether two independent samples of distributions are equal. Although the power of the Kolmogorov-Smirnov test statistic is not high, it is the standard test statistic for this general alternative. However, there are some advantages in using the Anderson-Darling test statistic instead of the Kolmogorov-Smirnov test statistic. We introduce two modified Anderson-Darling test statistics and list the exact critical values for small sample sizes. Moreover, we derive the limiting distribution of the proposed test statistic. Furthermore, we investigate the power performances of the proposed test statistic in various scenarios using Monte-Carlo simulations. Simulation studies and real data analysis reveal that the proposed test statistic is a good competitor to existing omnibus test statistics and has some advantages in certain situations.