Some notes on the location–scale Cucconi test

The best known and most used rank test for the location–scale problem is due to Lepage [Y. Lepage, A combination of Wilcoxon's and Ansari–Bradley's statistics, Biometrika 58 (1971), pp. 213–217.], but this paper is focused on the location–scale rank test of Cucconi [O. Cucconi, Un nuovo test non parametrico per il confronto tra due gruppi campionari, Giorn. Econom. XXVII (1968), pp. 225–248.], proposed earlier but not nearly as well-known. The test is of interest because, contrary to the other location–scale tests, it is not a quadratic form combining a test for location and a test for scale differences, and it is based on squared ranks and squared contrary-ranks. Moreover, it is easier to compute the test of Cucconi than those of Lepage, Manly–Francis, Büning–Thadewald, Neuhäuser, Büning and Murakami. Exact critical values for the test have been computed for the very first time. The power of the Cucconi test has been studied for the very first time and compared with that of the Lepage and other tests that include several Podgor–Gastwirth efficiency robust tests. Simulations show that the test of Cucconi maintains a size very close to α and is more powerful than the Lepage test, and therefore should be taken into account as a better alternative when it is not possible to develop an efficiency robust procedure for the problem at hand. The simulation study considers also the case of different shapes for the parent distributions, and the case of tied observations which is generally not considered in power studies. The presence of ties does not lower the performance of the Cucconi test, the contrary happens for the Lepage test. The tests are applied to real and fictitious biomedical data.