  The ultimate purpose of the statistical analysis of ordinal patterns is to
characterize the distribution of the features they induce. In particular,
knowing the joint distribution of the pair Entropy-Statistical Complexity for a
large class of time series models would allow statistical tests that are
unavailable to date. Working in this direction, we characterize the asymptotic
distribution of the empirical Shannon's Entropy for any model under which the
true normalized Entropy is neither zero nor one. We obtain the asymptotic
distribution from the Central Limit Theorem (assuming large time series), the
Multivariate Delta Method, and a third-order correction of its mean value. We
discuss the applicability of other results (exact, first-, and second-order
corrections) regarding their accuracy and numerical stability. Within a general
framework for building test statistics about Shannon's Entropy, we present a
bilateral test that verifies if there is enough evidence to reject the
hypothesis that two signals produce ordinal patterns with the same Shannon's
Entropy. We applied this bilateral test to the daily maximum temperature time
series from three cities (Dublin, Edinburgh, and Miami) and obtained sensible
results.
