Robustness and power of modified Lepage, Kolmogorov-Smirnov and Crame´r-von Mises two-sample tests

Abstract

For the two-sample problem with location and/or scale alternatives, as well as different shapes, several statistical tests are presented, such as of Kolmogorov-Smirnov and Cramér-von Mises type for the general alternative, and such as of Lepage type for location and scale alternatives. We compare these tests with the t-test and other location tests, such as the Welch test, and also the Levene test for scale. It turns out that there is, of course, no clear winner among the tests but, for symmetric distributions with the same shape, tests of Lepage type are the best ones whereas, for different shapes, Cramér-von Mises type tests are preferred. For extremely right-skewed distributions, a modification of the Kolmogorov-Smirnov test should be applied.

Hybrid Approach for Unbiased Coherence Estimation for Multitemporal InSAR

“…For a specific alternative hypothesis, there is no clear winner in test performance among all the nonparametric tests, and the efficiency of all the tests depends on the types of the underlying distributions. For example, for symmetric distributions with the same shape parameters, the Lepage-type tests are the best [48], [51], whereas for extremely right-skewed distributions, a modification of the KS test behaves better [48], [49]. Therefore, an "adaptive" procedure should be applied for selecting the most suitable test for a given data set.…”

“…Selector Statistics: Adaptive two-sample tests have been proposed for increasing the power of the test when the measurements are not normally distributed [37]- [41], [48]- [50]. For a specific alternative hypothesis, there is no clear winner in test performance among all the nonparametric tests, and the efficiency of all the tests depends on the types of the underlying distributions.…”

“…Following the concept of Hogg, we classify, at first, the type of the underlying distribution with respect to two measures, i.e., skewness and tail weight, and then select and carry out a test according to the identified distribution [38], [39], [41]. We choose the skewness and tail weight measures as [48]- [51] …”

A nonparametric location–scale statistic for detecting a change point

“…Yonetani [17][18][19] and Nakazawa and Rajendran [20] studied to detect the changes by using the Lepage test. After the Lepage test was proposed, various Lepagetype statistics have been proposed and discussed over the course of many years, e.g., Büning and Thaewald [21], Neuhäuser [22], Büning [23], Murakami [24]. One of the most famous and powerful Lepage-type statistic is a combination of the Wilcoxon and Mood [25] statistics, namely V, proposed by Pettitt [26].…”

Approximations to the distribution of a combination of the Wilcoxon and Mood statistics: A numerical comparison

“…After the Lepage statistic was proposed, various Lepage-type statistics have been proposed and discussed over the course of many years, e.g. Büning and Thaewald (2000), Neuhäuser (2000), Büning (2002), Murakami (2007). One of the most famous and powerful Lepage-type statistics is a combination of the Wilcoxon and Mood (1954) statistics, namely T , proposed by Pettitt (1976).…”