In this article, asynchronous distributed nonlinear estimation under directed networks is studied, where each agent updates its own estimate without waiting for other agents. The aim of each agent is to estimate the parameter of interest based on local, nonlinear and noisy measurements, together with neighbors' information received through network communication. To this end, a novel asynchronous distributed nonlinear estimator is proposed. By constructing an augmented system, it is rigorously proved that under some mild assumptions, the distributed estimator achieves consistent parameter estimates by selecting appropriate stepsizes. Furthermore, we also analyze the asymptotic property of the weighted estimate error sequence, including the asymptotic mean and asymptotic covariance. Finally, simulation results are given to verify the effectiveness of the devised algorithm.