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

Ferger, D.

Ann. Stat. 23, No. 5, 1848-1861 (1995).

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

Keywords:

maximizer of a two-sided random walk; consistency; nonstandard change-point problem; bootstrap; local power