## Nonparametric tests for nonstandard change-point problems.(English)Zbl 0843.62054

Summary: We consider independent random elements $$X_1,\dots, X_n$$, $$n \in N$$, with values in a measurable space $$({\mathcal H}, {\mathcal B})$$ so that $$X_1,\dots, X_{[n\theta]}$$ have a common distribution $$\nu_1$$ and the remaining $$X_{[n\theta] + 1},\dots, X_n$$ 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.

### MSC:

 62G10 Nonparametric hypothesis testing 62G20 Asymptotic properties of nonparametric inference 62G09 Nonparametric statistical resampling methods
Full Text: