Simulated power of the discrete Cramér-von Mises goodness-of-fit tests

The use of goodness-of-fit test statistics for discrete or categorical data is widespread throughout the research community with the Chi-Square the most popular when a researcher aims to determine if observed categorical data differs from a hypothesized multinomial distribution. Even for ordinal categorical data, the use of empirical distribution function (EDF) test statistics such as the Kolmogorov-Smirnov, the three Cramér-von Mises (A², W ² and U² as defined below) and various modifications of these are limited in the literature. Power studies of the EDF type test statistics are even more limited. This paper compares the simulated power of the three Cramér-von Mises test statistics with that of the Chi-Square test statistic for a uniform null hypothesis against a variety of alternative distributions which are summarized in Figure 1. Recommendations are made on which is the most powerful test statistic for the predefined alternative distributions. (Figure Presented) The results of the simulated power studies in this paper lead to the following general recommendations: • For trend type alternatives A² and W² appear much more powerful than U² and χ². (See Figure 2 for a uniform null against a decreasing trend alternative distribution). • For all the other investigated alternative distributions U² and χ² appear much more powerful than A² and W². (See Figure 3 for a uniform null against a leptokurtic type alternative distribution). (Graph Presented).