A test when the Fisher information may be infinite, exemplified by a test for marginal independence in extreme value distributions

For testing independence between the components of the multivariate logistic extreme value distribution, the Fisher Information is infinite and the score statistic converges only at the rate of O(n−1/2logn). Consequently, the asymptotic optimality properties of the usual likelihood based methods do not necessarily carry over to this case. Motivated by this specific problem, this paper develops a general method based on the Cramér–von Mises discrepancy. This method does not require finite Fisher information. It is shown that this test statistic converges at the regular rate of O(n−1/2)O(n−1/2). The test statistic has a closed form and it can be computed easily without using an iterative method or numerical integration. The asymptotic critical values can be obtained from a table for chi-square distribution. In a simulation study involving the test of marginal independence in bivariate extreme value distributions, the test proposed in this paper performed better than its main competing ones. While the method was motivated by, and developed for, a particular topic in inference for multivariate extreme value distributions, it is applicable more broadly than just for inference in multivariate extreme value distributions.