Abstract
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. The RWS applies to various types of parametric, nonparametric and semiparametric estimators from likelihood, quasi-likelihood and least squares. The consistency and asymptotic normality of the proposed renewable estimator are established, and the oracle property is obtained. Moreover, these properties are always satisfied, without the familiar constraints on the number of data batches, which means that the new method is adaptive to the situation where streaming data sets arrive perpetually. The performance of the method is further illustrated by various numerical examples from simulation experiments and real data analysis.
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