  Entropy production plays an important role in the regulation and stability of
active matter systems, and its rate quantifies the nonequilibrium nature of
these systems. However, entropy production is hard to be experimentally
estimated even in some simple active systems like molecular motors or bacteria,
which may be modeled by the run-and-tumble particle (RTP), a representative
model in the study of active matters. Here we resolve this problem for an
asymmetric RTP in one-dimension, firstly constructing a finite time
thermodynamic uncertainty relation (TUR) for a RTP, which works well in the
short observation time regime for entropy production estimation. Nevertheless,
when the activity dominates,i.e., the RTP is far from equilibrium, the lower
bound for entropy production from TUR turns to be trivial. We address this
issue by introducing a recently proposed high-order thermodynamic uncertainty
relation (HTUR), in which the cumulant generating function of current serve as
a key ingredient. To exploit the HTUR, we adopt a novel method to analytically
obtain the cumulant generating function of the current we study, with no need
to explicitly know the time-dependent probability distribution. The HTUR is
demonstrated to be able to estimate the steady state energy dissipation rate
accurately because the cumulant generating function covers higher-order
statistics of the current, including rare and large fluctuations besides its
variance. Compared to the conventional TUR, the HTUR could give significantly
improved estimation of energy dissipation, which can work well even in the
far-from equilibrium regime. We also provide a strategy based on the improved
bound to estimate the entropy production from moderate amount of trajectory
data for experimental feasibility.
