Second Order Hadamard Differentiability in Statistical Applications

A formulation of the second-order Hadamard differentiability of (extended) statistical functionals and some related theoretical results are established. These results are applied to derive the limiting distributions of a class of generalized Cramér–von Mises type test statistics, which include some proposed new ones for the tests of goodness of fit in the 3-sample problems, the tests in linear regression models, and the tests of bivariate independence, as special cases.