We consider a real Gaussian process X having a global unknown smoothness
(r0,β0),
r_{\scriptscriptstyle 0}\in \mathds{N}_0 and β0∈]0,1[, with X(r0) (the mean-square derivative of
X if r0≥1) supposed to be locally stationary with
index β0. From the behavior of quadratic variations
built on divided differences of X, we derive an estimator of
(r0,β0) based on - not
necessarily equally spaced - observations of X. Various numerical studies of
these estimators exhibit their properties for finite sample size and different
types of processes, and are also completed by two examples of application to
real data.Comment: 28 page