  This paper establishes unified frameworks of renewable weighted sums (RWS)
for various online updating estimations in the models with streaming data sets.
The newly defined RWS lays the foundation of online updating likelihood, online
updating loss function, online updating estimating equation and so on. The idea
of RWS is intuitive and heuristic, and the algorithm is computationally simple.
This paper chooses nonparametric model as an exemplary setting. The RWS applies
to various types of nonparametric estimators, which include but are not limited
to nonparametric likelihood, quasi-likelihood and least squares. Furthermore,
the method and the theory can be extended into the models with both parameter
and nonparametric function. The estimation consistency and asymptotic normality
of the proposed renewable estimator are established, and the oracle property is
obtained. Moreover, these properties are always satisfied, without any
constraint on the number of data batches, which means that the new method is
adaptive to the situation where streaming data sets arrive perpetually. The
behavior of the method is further illustrated by various numerical examples
from simulation experiments and real data analysis.
