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.