Equivalence between Mallows and Girone-Cifarelli tests

In this article, it is shown that the Mallows goodness-of-fit test is equivalent to the Girone-Cifarelli test. Models for which the Mallows and Girone-Cifarelli are more powerful than the Kolmogorov-Smirnov test (commonly used) are described in Friedrich and Trede (1996) and Borroni and Chiodini (2013) respectively. Our hope is that this paper will help promote the use of these appealing tests based on the L1 distance. Particularly the use of the Girone-Cifarelli test which is computationally more efficient than the Mallows test.