  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. Different sensors can have different
step sizes and triggering thresholds, enabling the parameter estimation process
to be conducted in a fully distributed way. We prove that the proposed
algorithm has mean-square and almost-sure convergence respectively, under
proper conditions of network connectivity and system collective observability.
The collective observability is the possibly mildest condition, since it is a
spatially and temporally collective condition of all sensors and allows sensor
observation matrices to be time-varying, stochastic, and non-stationary.
Moreover, we provide estimates for the convergence rates, which are related to
the step sizes as well as the triggering thresholds. Furthermore, we prove that
the communication rate is decaying to zero with a certain rate almost surely as
time goes to infinity. We show that it is feasible to tune the thresholds and
the step sizes such that requirements of algorithm convergence and
communication rate decay are satisfied simultaneously.Numerical simulations are
provided to illustrate the developed results.
