  The statistical analysis of data stemming from dynamical systems, including,
but not limited to, time series, routinely relies on the estimation of
information theoretical quantities, most notably Shannon entropy. To this
purpose, possibly the most widespread tool is provided by the so-called plug-in
estimator, whose statistical properties in terms of bias and variance were
investigated since the first decade after the publication of Shannon's seminal
works. In the case of an underlying multinomial distribution, while the bias
can be evaluated by knowing support and data set size, variance is far more
elusive. The aim of the present work is to investigate, in the multinomial
case, the statistical properties of an estimator of a parameter that describes
the variance of the plug-in estimator of Shannon entropy. We then exactly
determine the probability distributions that maximize that parameter. The
results presented here allow to set upper limits to the uncertainty of entropy
assessments under the hypothesis of memoryless underlying stochastic processes.
