Nonparametric tests for nonstandard change-point problems

We consider independent random elements X(1),...,X(n), n is an element of N, with values in a measurable space (L, B) so that X(1),...,X([n0]) have a common distribution nu(1) and the remaining X([n0]+1),...,X(n) have a common distribution nu(2) not equal v(1), for some theta is an element of (0, 1). The change point theta as well as the distributions are unknown. A family of tests is introduced for the nonstandard change-point problem H-0: theta is an element of Theta(0) versus H-1: theta is not an element of Theta(0), where Theta(0) is an arbitrary subset of (0, 1). The tests are shown to be asymptotic level-alpha tests and to be consistent on a large class of alternatives. The same holds for the corresponding bootstrap versions of the tests. Moreover, we present a detailed investigation of the local power.