A modified Cucconi test for location and scale change alternatives

The familiar approach to develop a test for jointly detecting location and scale changes is to combine a test for location and a test for scale. For the same problem, the test of Cucconi should be considered because it is different than the other tests being based on the squares of ranks and contrary-ranks. It has been previously shown that the Cucconi test is robust in level and is more powerful than the Lepage test which is the most familiar test for the location-scale problem. A modification of the Cucconi test is proposed. The idea is to modify this test consistently with the familiar approach which develops a location-scale test by combining a test for location and a test for scale. More precisely, we will combine the Cucconi test with the Wilcoxon rank test for location and a modified Levene test following the theory of the nonparametric combination. A power comparison of this modified Cucconi test with the original one, the Lepage test and the Podgor-Gastwirth $PG2$ test shows that the modified Cucconi test is robust in size and markedly more powerful than the other tests for every considered type of distributions, from short- to normal- and long-tailed ones. A real data example is discussed