Testing for the Presence of Self‐Similarity of Gaussian Time Series Having Stationary Increments

A method for testing for the presence of self‐similarity of a Gaussian time series with stationary increments is presented. The test is based on estimation of the distance between the time series and a set of time series containing all the fractional Brownian motions. This distance is constructed from two estimations of multiscale generalized quadratic variations expectations. The second one requires regression estimates of the self‐similarity index H. Two estimations of H are then introduced. They present good robustness and computing time properties compared with the Whittle approach, with nearly similar convergence rate. The test is applied on simulated and real data. The self‐similarity assumption is notably accepted for the famous Nile River data.