## The Annals of Statistics

- Ann. Statist.
- Volume 23, Number 5 (1995), 1848-1861.

### 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.

#### Article information

**Source**

Ann. Statist., Volume 23, Number 5 (1995), 1848-1861.

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176324326

**Digital Object Identifier**

doi:10.1214/aos/1176324326

**Mathematical Reviews number (MathSciNet)**

MR1370310

**Zentralblatt MATH identifier**

0843.62054

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G10: Hypothesis testing

Secondary: 62G20: Asymptotic properties 62G09: Resampling methods 60F05: Central limit and other weak theorems

**Keywords**

Tests for change-point problems maximizer of a two-sided random walk consistency local power bootstrap

#### Citation

Ferger, D. Nonparametric Tests for Nonstandard Change-Point Problems. Ann. Statist. 23 (1995), no. 5, 1848--1861. doi:10.1214/aos/1176324326. https://projecteuclid.org/euclid.aos/1176324326