  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) for d $\in$ (--0.5, 0.5) can
be extended to a semiparametric class of Gaussian fractionally integrated
processes with memory parameter d $\in$ (--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 \textless{} 0.5 and d $\ge$ 0.5, as well as a
fractional unit roots test distinguishing the case d = 1 from the case d
\textless{} 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. Keywords: Gaussian
fractionally integrated processes; semiparametric estimators of the memory
parameter; test of long-memory; stationarity test; fractional unit roots test.
