  We study distributed estimation of a high-dimensional static parameter vector
through a group of sensors whose communication network is modeled by a fixed
directed graph. Different from existing time-triggered communication schemes,
an event-triggered asynchronous scheme is investigated in order to reduce
communication while preserving estimation convergence. A distributed estimation
algorithm with a single step size is first proposed based on an event-triggered
communication scheme with a time-dependent decaying threshold. With the
event-triggered scheme, each sensor sends its estimate to neighbor sensors only
when the difference between the current estimate and the last sent-out estimate
is larger than the triggering threshold. We prove that the proposed algorithm
has mean-square and almost-sure convergence respectively, under an integrated
condition of sensor network topology and sensor measurement matrices. The
condition is satisfied if the topology is a balanced digraph containing a
spanning tree and the system is collectively observable. Moreover, we provide
estimates for the convergence rates, which are related to the step size as well
as the triggering threshold. Furthermore, as an essential metric of sensor
communication intensity in the event-triggered distributed algorithms, the
communication rate is proved to decay to zero with a certain speed almost
surely as time goes to infinity. We show that given the step size, adjusting
the decay speed of the triggering threshold can lead to a tradeoff between the
convergence rate of the estimation error and the decay speed of the
communication rate. Specifically, increasing the decay speed of the threshold
would make the communication rate decay faster, but reduce the convergence rate
of the estimation error. Numerical simulations are provided to illustrate the
developed results.
