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 Cramér-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.
Highlights
Over the last two decades various authors have studied the change-point problem under long-range dependence and classical methods are often found to yield different results than under short-range dependence
The CUSUM test is studied in Csorgoand Horvath (1997) and compared to the Wilcoxon change-point test in Dehling et al (2012)
Ling (2007) investigates a Darling-Erdos-type result for a parametric change-point test, and estimators for the time of change are considered in Horvath and Kokoszka (1997) and Hariz et al (2009)
Summary
Over the last two decades various authors have studied the change-point problem under long-range dependence and classical methods are often found to yield different results than under short-range dependence. For stationary sequences that exhibit long-range dependence, Dehling and Taqqu (1989a) proved that the limit process is of the form {J(x)Z(t)}t,x, where J(x) is a deterministic function and the process is called semi-degenerate. A similar limit structure was later obtained independently by Ho and Hsing (1996) and Giraitis et al (1996a) for long-range dependent moving-average sequences It is the main goal of this paper to derive the limit distribution of changepoint statistics of the type (1.1) and (1.2) under local alternatives. We apply these results to derive the asymptotic relative efficiency (ARE) of several change-point tests.
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