  In this work we implement the so-called matching time estimators for
estimating the entropy rate as well as the entropy production rate for symbolic
sequences. These estimators are based on recurrence properties of the system,
which have been shown to be appropriate to test irreversibility specially when
the sequences have large correlations or memory. Based on limit theorems for
matching-times we derive a maximum likelihood estimator for entropy rate
assuming that we have a set of moderately short symbolic time-series of finite
random duration. We show that the proposed estimator has several properties
that makes it adequate to estimate entropy rate and entropy production rate (or
to test irreversibility) when the sample sequences have different lengths such
as the coding sequences of DNA. We test our approach in some controlled
examples of Markov chains. We also implement our estimators in genomic
sequences to show that the degree of irreversibility coding sequences of human
DNA is significantly larger than the corresponding non-coding sequences.
