Nonparametric Tests for Nonstandard Change-Point Problems

Abstract

We consider independent random elements $X_1, \ldots, X_n, n \in \mathbb{N}$, with values in a measurable space $(\mathscr{X}, \mathscr{B})$ so that $X_1, \ldots, X_{\lbrack n\theta\rbrack}$ have a common distribution $\nu_1$ and the remaining $X_{\lbrack n\theta\rbrack + 1}, \ldots, X_n$ have a common distribution $\nu_2 \neq \nu_1$, for some $\theta \in (0, 1)$. The change point $\theta$ as well as the distributions are unknown. A family of tests is introduced for the nonstandard change-point problem $H_0: \theta \in \Theta_0$ versus $H_1: \theta \not\in \Theta_0$, 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.