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Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic

The increment ratio (IR) statistic was first defined and studied in Surgailis {\it et al.} (2007) for estimating the memory parameter either of a stationary or an increment stationary Gaussian process. Here three extensions are proposed in the case of stationary processes. Firstly, a multidimensional central limit theorem is established for a vector composed by several IR statistics. Secondly, a goodness-of-fit $χ^2$-type test can be deduced from this theorem. Finally, this theorem allows to construct adaptive versions of the estimator and test which are studied in a general semiparametric frame. The adaptive estimator of the long-memory parameter is proved to follow an oracle property. Simulations attest of the interesting accuracies and robustness of the estimator and test, even in the non Gaussian case.