Abstract In this paper, we show that the central limit theorem (CLT) satisfied by the data-driven Multidimensional Increment Ratio (MIR) estimator of the memory parameter d established in Bardet and Dola (2012. Adaptive Estimator of the Memory Parameter and Goodness-of-Fit Test Using a Multidimensional Increment Ratio Statistic.” Journal of Multivariate Analysis 105:222–40) for dϵ(–0.5, 0.5) can be extended to a semiparametric class of Gaussian fractionally integrated processes with memory parameter dϵ(–0.5, 1.25). Since the asymptotic variance of this CLT can be estimated, by data-driven MIR tests for the two cases of stationarity and non-stationarity, so two tests are constructed distinguishing the hypothesis d < 0.5 and d≥0.5, as well as a fractional unit roots test distinguishing the case d=1 from the case d < 1. Simulations done on numerous kinds of short-memory, long-memory and non-stationary processes, show both the high accuracy and robustness of this MIR estimator compared to those of usual semiparametric estimators. They also attest of the reasonable efficiency of MIR tests compared to other usual stationarity tests or fractional unit roots tests.

中文翻译：

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基于数据驱动的多维增量比率统计的半参数平稳性和分数单位根检验

摘要本文表明，在Bardet和Dola（2012.内存参数和优度的自适应估计器）中建立的数据驱动多维参数多维增量比（MIR）估计器满足的中心极限定理（CLT）。多维度分析杂志105：222–40）的dϵ（–0.5，0.5）可以扩展为半参数类的高斯分数积分过程，其存储参数为dϵ（–0.5，0.5）。 1.25）。由于此CLT的渐近方差可以通过平稳性和非平稳性两种情况的数据驱动MIR检验来估计，因此构造了两个检验以区分假设d <0.5和d≥0.5以及分数单位进行根检验，以区分d = 1和d <1。对多种短期，长期和非平稳过程进行的仿真显示，与常规的半参数估计器相比，此MIR估计器具有很高的准确性和鲁棒性。他们还证明了与其他常用的平稳性测试或分数单位根测试相比，MIR测试的合理效率。