Change-point tests under local alternatives for long-range dependent processes

TL;DR

This paper studies change-point detection in long-range dependent Gaussian processes, analyzing the asymptotic behavior of various tests under local alternatives and demonstrating their efficiency and finite sample performance.

Contribution

It provides new asymptotic distributions for change-point tests in long-range dependent processes and compares their efficiencies, including the case of changing Hermite rank.

Findings

Asymptotic distributions of tests are derived under local alternatives.

The tests have an asymptotic relative efficiency of 1 in mean-shift scenarios.

Finite sample simulations confirm the theoretical results.

Abstract

We consider the change-point problem for the marginal distribution of subordinated Gaussian processes that exhibit long-range dependence. The asymptotic distributions of Kolmogorov-Smirnov- and Cram\'{e}r-von Mises type statistics are investigated under local alternatives. By doing so we are able to compute the asymptotic relative efficiency of the mentioned tests and the CUSUM test. In the special case of a mean-shift in Gaussian data it is always $1$. Moreover our theory covers the scenario where the Hermite rank of the underlying process changes. In a small simulation study we show that the theoretical findings carry over to the finite sample performance of the tests.