  For equidistant discretizations of fractional Brownian motion (fBm), the
probabilities of ordinal patterns of order d=2 are monotonically related to the
Hurst parameter H. By plugging the sample relative frequency of those patterns
indicating changes between up and down into the monotonic relation to H, one
obtains the Zero Crossing (ZC) estimator of the Hurst parameter which has found
considerable attention in mathematical and applied research.
  In this paper, we generally discuss the estimation of ordinal pattern
probabilities in fBm. As it turns out, according to the sufficiency principle,
for ordinal patterns of order d=2 any reasonable estimator is an affine
functional of the sample relative frequency of changes. We establish strong
consistency of the estimators and show them to be asymptotically normal for
H<3/4. Further, we derive confidence intervals for the Hurst parameter.
Simulation studies show that the ZC estimator has larger variance but less bias
than the HEAF estimator of the Hurst parameter.
