The Increment Ratio statistic under deterministic trends

  • K. Bružaitė
  • M. Vaičiulis


The Increment Ratio (IR) statistic (see (1.1) below) was introduced in Surgailis et al. [16]. The IR statistic can be used for testing nonparametric hypotheses for d-integrated (−1/2 < d < 5/4) behavior of time series, including short memory (d = 0), (stationary) long-memory (0 < d < 1/2), and unit roots (d = 1). For stationary/stationary increment Gaussian observations, in [16], a rate of decay of the bias of the IR statistic and a central limit theorem are obtained. In this paper, we study the asymptotic distribution of the IR statistic under the model X t = X t 0 + g N(t) (t = 1, …, N), where X t 0 is a stationary/stationary increment Gaussian process as in [16], and g N(t) is a slowly varying deterministic trend. In particular, we obtain sufficient conditions on X t 0 and g N(t) under which the IR test has the same asymptotic confidence intervals as in the absence of the trend. We also discuss the asymptotic distribution of the IR statistic under change-points in mean and scale parameters.


central limit theorem increment ratio statistic fractional Brownian motion Gaussian processes long memory 


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Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsVilniusLithuania

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