Hermite series based distribution function estimators have recently been applied in the context of sequential quantile estimation. These distribution function estimators are particularly useful because they allow the online (sequential) estimation of the full cumulative distribution function. This is in contrast to the empirical distribution function estimator and smooth kernel distribution function estimator which only allow sequential cumulative probability estimation at particular values on the support of the associated density function. Hermite series based distribution function estimators are well suited to the settings of streaming data, one-pass analysis of massive data sets and decentralised estimation. In this article we study these estimators in a more general context, thereby redressing a gap in the literature. In particular, we derive new asymptotic consistency results in the mean squared error, mean integrated squared error and almost sure sense. We also present novel Bias-robustness results for these estimators. Finally, we study the finite sample performance of the Hermite series based estimators through a real data example and simulation study. Our results indicate that in the general (non-sequential) context, the Hermite series based distribution function estimators are inferior to smooth kernel distribution function estimators, but may remain compelling in the context of sequential estimation of the full distribution function.