  Offline estimators are often inadequate for real-time applications.
Nevertheless, many online estimators are found by sacrificing some statistical
efficiency. This paper presents a general framework to understand and construct
efficient nonparametric estimators for online problems. Statistically, we
choose long-run variance as an exemplary estimand and derive the first set of
sufficient conditions for O(1)-time or O(1)-space update, which allows
methodological generation of estimators. Our asymptotic theory shows that the
generated estimators dominate existing alternatives. Computationally, we
introduce mini-batch estimation to accelerate online estimators for real-time
applications. Implementation issues such as automatic optimal parameters
selection are discussed. Practically, we demonstrate how to use our framework
with recent development in change point detection, causal inference, and
stochastic approximation. We also illustrate the strength of our estimators in
some classical problems such as Markov chain Monte Carlo convergence diagnosis
and confidence interval construction.
