Some Notes on the Cucconi Rank Test for Location-Scale Problems

In the paper, the location-scale problem is addressed within the rank testing framework. The best known and most used rank test for the location-scale problem is the Lepage test. There exist another rank test for the same problem which is earlier, but not known outside the Italian statistical school, and no power study has ever been done on it: the Cucconi test. We show that this test performs like the Lepage one and sometimes it works better. Moreover, it is always more conservative. Therefore the Cucconi test should be taken into account as an alternative to the Lepage test. We also show that it is worth to use, for the scale problem, a location-scale rank test like Lepage or Cucconi instead of the Ansari-Bradley rank test. In fact, we noted that when locations are equal the power loss is generally not much (in particular for the Cucconi test which sometimes is even more powerful than A), and that when locations are unequal the A test may be wholly useless in detecting scale shifts.