The Increment Ratio statistic under deterministic trends

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 _t⁰ + g _N(t) (t = 1, …, N), where X _t⁰ 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 _t⁰ 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. Partially supported by the bilateral France-Lithuania scientific project Gilibert and Lithuanian State Science and Studies Foundation, grant No. T-25/08.